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CAC Document No. 17
A FAST FOURIER TRANSFORM SUBROUTINE
FOR ILLIAC IV
by
James E. Stevens, Jr.
October 15, 1971
BIBLIOGRAPHIC VTA 1'• Kcporj No.
SHEET | UIUC-CAC-DN-71-172. 3. Recipient's Accession No.
4. I i! le and >ul-t it le
A FA3T FOURIER TRANSFORM SUBROUTINE FOR ILL IAC IV
5. Report 1 '.Me |
October 15, 19716. |
7. Author(s)
James E. Stevens, -Jr.
8. Performing Organization Kept.No.
CAC _179. Performing Organization Name and Address
Center for Advanced ComputationUniversity of Illinois at Urbana-ChampaignUrbana, Illinois 61801
10. Project/ I'ask/Work Unit No.
11. Contract Grant No.
DAHC0U-72-C-0001
12. Sponsoring Organization Name and Address
U. S. Army Research Office-DurhamDuke StationDurham, North Carolina
13. Type ot Report & PeriodCovered
Fe search14.
15. Supplementary Notes
16. Abstracts rQ^ report i s a description of a Fast Fourier Transform (FFT)subroutine written in assembly language for the ILLIAC IV computer. Thesubroutine uses the Cooley-Tukey algorithm for performing discrete Fouriertransforms. The parallel nature of the Cooley-Tukey method lends itself verywell to a highly parallel machine like ILLIAC IV. Timing- simulation results haveshown that this program will perform Fourier transforms faster than they are
being done on any existing computer system.
1
1
i
17. Key Uords and Document Analysis. 17o. Descriptors
Seismic Data ProcessingASKFast Fourier Transform (FFT)Discrete Fourier Transform (DFT)
Time SeriesFrequency Spectrum
17b. Idcntif icrs/Open-Ended Terms
17c. COSATI Field/Group
18. Avail.ibi lity St .it< memNo restriction on distribution.
Available from National Technical Information
Service, Springfield, Virginia 22151
19. Security ( lass (ThisReport)
IWI.ASSIFM-n
21. No. ot Pages
6720. Security ( lass (This
PageUNC1.ASSIEI1 n
22. Price
FOHM NTIli-15 IHI.V. J- / J I
Tins FORM MAY DIE REPRODUCED UjCOMM-DC I 4 5 2 P 7 2
Digitized by the Internet Archive
in 2012 with funding from
University of Illinois Urbana-Champaign
http://archive.org/details/fastfouriertransOOstev
CAC Document No. 17
A FAST FOURIER TRANSFORM SUBROUTINE
FOR ILLIAC IV
by
James E. Stevens, Jr.
Center for Advanced ComputationUniversity of Illinois at Urbana-Champaign
Urbana, Illinois 61801
October 15, 1971
This work was supported in part by the Advanced Research ProjectsAgency of the Department of Defense and was monitored by the
U.S. Army Research Office-Durham under Contract No. DAHC0I4 72-C-0001.
INGWEEWNG U»*»~
ABSTRACT
This report is a description of a Fast Fourier Transform (FFT)
subroutine written in assembly language for the ILLIAC IV computer. The
subroutine uses the Cooley-Tukey algorithm for performing discrete Fourier
transforms. The parallel nature of the Cooley-Tukey method lends itself
very well to a highly parallel machine like ILLIAC IV. Timing simulation
results have shown that this program will perform Fourier transforms faster
than they are being done on any existing computer system.
TABLE OF CONTENTS
Introduction 1
Subroutine Specifications 2
Implementation of the Cooley-Tukey Procedure 2
Scrambling 8
Future Considerations 15
ILLIAC IV Fast Fourier Transform Users Manual 17
APPENDIX A
APPENDIX B
APPENDIX C
APPENDIX D
References
Standard Subroutine Linking 19
Demonstration Run 20
Timing Summary 28
Subroutine Listing 29
63
Introduction
The Fourier transform is one of the most important mathematical
aids to signal processing. The basic property of the Fourier transform is
its ability to distinguish waves of different frequencies which have been
additively combined. The result of a Fourier transform on a complex func-
tion of time is a frequency spectrum of that function. This resulting fre-
quency spectrum is also a complex function. To facilitate processing on
digital machines, a method for transforming discrete functions of time into
discrete functions of frequency has been derived. The time function must
consist of evenly spaced samples of some interval^ T. A function of N samples
could be represented as f(nT) for < n < N - 1. The spectrum of the func-
tion would also consist of discrete values evenly spaced in frequency. If
the spectrum were composed of N points, the interval of frequency, q, would
be 2tt/NT. The spectrum F(koJ, < k < N - 1 would be given by
N-l
F(koJ = Z f(nT)e"jnTkn
(l)
n=o
This equation defines the discrete Fourier transform (DFT) of a
sequence of N samples. It follows directly that the DFT is linear since
DFT[f(nT) + g(nT)} = DFT{f(nT)} + DFT{g(nT)} (2)
and
DFT(c(f(nT))) = c[ DFT{f(nT)}
]
(3)
This follows quickly from the definition.
There exists an inverse DFT, a transform which maps a discrete
Fourier transform back into the sequence from which it was composed. It is
given by
f(nT) .| lfo FdB).*"" W
-1-
which differs from the DFT equation 1, only by a scale factor and by the
sign of the exponential. The same procedure used for the DFT can thus be
applied to a frequency spectrum to perform the inverse Fourier Transform by
simply dividing the result by N.
2The above method for computing DFT requires N complex operations
to be performed in order to transform a function with N samples. The method
of Cooley and Tukey [1] eliminates redundant steps in that method resulting
in a procedure requiring only N logp
N complex operations. This method is
highly parallel and therefore very well suited for use on ILLIAC IV [2].
Subroutine Specifications
The ILLIAC IV fast Fourier subroutine accepts discrete input func-
tions composed of N complex samples where N is an even power of two between
8 and U096, inclusive. The output of the subroutine is a discrete frequency
spectrum composed of N complex samples as defined by equation (l). The out-
put is stored over the input in ascending sequence in a contiguous block of
memory. N words of storage are used, where each word contains the real
(outer) and imaginary (inner) portions of one sample in the 32-bit format.
Input to the subroutine must also be stored in this manner; therefore, all
values have 2^-bit significance and are represented in floating point.
Several functions can be input to the subroutine at one time as long as
they are stored contiguously and provided that they all are the same length.
If the transform of a function of M samples is desired where M is not a
power of two, N-M zero samples should be added to the function where N is
the smallest power of two greater than M. The result can be input to the
subroutine as a function of N samples. The output will be a frequency
spectrum equivalent to the transform of the original function of M samples
with a smaller frequency spacing.
Implementation of the Cooley-Tukey Procedure
The method of Cooley and Tukey [1] is illustrated in Figure 1.
Implementation of this method on ILLIAC IV has been discussed by G.M. Ackins
[3]- Ackins' methods have been refined and generalized for use in this sub-
routine. The following description gives the details of the final imple-
mentation.-2-
vw\
Figure 1
In Figure 1, f represents f(nT) and F represents F(npJ. Also W
.2
is used to replace the constant e N. At each node in the diagram, a com-
plex multiply involving the lower of the two inputs to the mode and the
constant, W, raised to the appropriate power is performed. The result is
added to the upper of the two inputs giving an answer which is used as an
-3-
input to the next iteration until the transform is completed. At each stage
in the process where an iteration is complete, the partial results are
stored into the initial locations so that the data is transformed in place.
Figure \ illustrates how this transform would be accomplished in parallel
on an eight PE ILLIAC IV.
Initial
_data_
f
First Second Third
iteration iteration iteration
Figure 2
-h-
In Figure 2, the arrows represent routing of data between PE's
while the nodes represent the operations which take place within the PE's
as given for Figure 1. f(t), and all of the necessary powers of W are
stored in PE memory for use in the calculation. From Figures 1 and 2, it
is seen that the input data has been "shuffled" so that the output appears
in ascending sequence. This shuffling also causes the routing (arrow)
pattern to be very regular. The algorithm for ordering input data is ex-
plained in a later section of this report which describes the routine
called Scramble which performs the shuffling in ILLIAC IV.
The method described above and illustrated in Figure 2 can be
generalized to transform much larger functions. Careful examination of
Figure 1 reveals that after two iterations, two separate four point trans-
forms have been completed. A four point transform is depicted in Figure 3«
It should be noted that the value of W depends on the size of the function
being transformed.
^ &Z. /V^\y ^ 1
/ G -,w W2
/ >/ G3
w2 u 3
W
Figure 3
The eight point transform can now be represented in terms of the
four point transform as depicted in Figure h.
-5-
go
V>
g2
o
DISCRETE
FOURIER
TRANSFORM(N=4)
DISCRETEFOURIER
TRANSFORM(N=4)
Figure h
-6-
The equality of these two representations (Figure 1 and Figure k)
is important to the structure of the subroutine. In an eight point transform,
_.2 ^ _. _.2W = e B, so that W = e . In a four point transform, W = e ^, so that
W - e . Thus, an eight point transform can he accomplished by performing
two four point transforms and iterating once more. In the general case
any N point transform, where N is a power of two, can he accomplished by
performing two N/2 point transforms and combining those results with one
final iteration. One consequence of this is that the first step in per-
forming any size transform is to perform an eight point transform on each
group of eight points. The entire subroutine is built up from that basis.
The other consequence of this partitioning of the procedure is
that each stage is completely independent. When a function composed of
128 data samples is to be transformed on ILLIAC IV, it is simply stored in
two rows of PE memory. One row of PE memory consists of one word in each
PE. The first 6k samples are stored in one row (after proper scrambling)
and the second half are stored in the other row. This is referred to as
"folding" 128 words of data into two 6k word rows of memory. A 6k point
transform can now easily be performed on the first half of the data since
this operation is independent of the other half. This 6k point transform
is a simple extension of the eight point transform of Figure 2, using 6k
PE's. Another independent 6k point transform is then performed on the
second half of the data.
The results of these two 6k point transforms can be combined in
one simple iteration giving the final 128 point transform. Since the in-
put data was folded in PE memory, no routing is required to align the two
halves for this final iteration simplifying it even further. This can be
illustrated by observing that the final iteration of each transform re-
quires a route of N/2 positions. For a 128 point transform this would be
a route of 6k which is equal to a route of in a single quadrant ILLIAC
IV. Furthermore, every transform larger than 6k requires a final route
which is a multiple of 6k which means no routing is actually required.
Since there is a smaller proportion of time spent in routing, the largest
transform is actually the most efficient in its computation.
-7-
The final generalization is that each larger 'transform above 6k
is simply accomplished by folding the function and treating the halves in-
dependently until the final iteration. The size of the function will de-
termine how many eight point transforms are done as the first step. Also
the number of functions to be transformed at a time can be taken into
account through proper loop control. For example, two 20U8 point transforms
could quite easily be done at the same time. They would each be scrambled
separately and then the transform would proceed as if the two functions com-
posed a single U096 point function—only, the final iteration would not be
performed.
Scrambling
The shuffling of data in the processing elements before perform-
ing the transform allows the computation to be done in place, that is, by
writing all intermediate results over the original data. This process will
be referred to as prescrambling and when it is applied the transform gen-
erates results which are sequentially ordered across PE's. The original
function is composed of discrete samples at even spaced time intervals.
They are assigned indices through N in order. Each index can be rep-
resented by a binary number with log N digits, KnK . . . K , where
n = log N - 1. The scrambling process requires that each sample having
index KAK. . . . K be interchanged with the sample having index K K , ... K^.1 n c n n-1
Thus by a simple "bit reversal" of each binary index the destination of each
sample in the scrambling process is determined. Note that in Figure 1 a
three digit bit reversal on the sequence 0,1,2,3A>5>6>7 yields the proper
sequence for input to the transform procedure, i.e., 0,^,2,6,1,5,3,7- Con-
sider the following example to see how the scrambling is accomplished within
ILLIAC IV. The input data is represented by indices only.
-8-
PEo
Initialfunction 1 2 3 k 5 6 7 8 9 10 11 12 13
Bit reversedindices 8 k 12 2 10 6 lU 1 9 5 13 3 11
1stiteration 1 2 3 U 5 6 7 8 9 10 11 12 13
Route (7) * * t T~* t *
2nditeration 8
Route (2)
Properlyordered 8
PE15
lk 15
7 15
lk 15
2 10 J+ 12 6 li+ l 9 3 11 5 13 7 15
k 2 2 10 6 lU 1 9 5 13 3 11 7 15
This process can be generalized for scrambling N points in the
following manner. The odd numbered PE's less than N/2 send data to the
corresponding even numbered PE's greater than w/2 and receive data back
from them. The next iteration is the same except the first N/2 PE's and
the second N/2 PE's perform it independently and handle the data by pairs.
Note that only two routes are required for each iteration. By this method
ILLIAC IV can scramble a 6k point data set in three iterations requiring
only 6 routes. For larger data sets the data must be folded in storage and
a different scheme is used. The following example shows how a sixteen point
data set would be scrambled by a k PE ILLIAC IV.
PEo
PE PE2
PE3
L0C 1 2 3
L0C+1 k 5 6 7
LOC+2 8 9 10 11
L0C+3 12 13 lk 15
Proper reorderingdetermined by bit reversal
PEo
PE PE2
PE3
8 k 12
2 10 6 11+
1 9 5 13
3 11 7 15
-9-
The underlined indices denote the positions that are unaffected
by scrambling since data in these positions does not need to be moved. The
location of these stationary positions within each PE is given by perform-
ing a bit reverse operation on the PE numbers. Each PE now loads a register
with data from the location corresponding to the stationary location in the
next higher numbered PE.
Register
Memory
L0C
Lyic+i
L0C+2
LOC+3
8..
^11+> 3>
PEo
PE1
PE2
PE3
k
12
1
5'
9
13
2
6
10
Ik"
3J
7
11
15
Each PE routes the data in its register to the next higher
numbered PE. The rightmost PE routes back to the first PE.
PEQ
PE PE2
PE
register Ik
K7—o—<cr
PEQ
PE1
PE2
PE3
register 3 8 5 Ik
Now the data in each register is exchanged with the data from the
location in the PE corresponding to the stationary location in the next
lower numbered PE.
-10-
PEQ
PE1
PE2
PE
Register
Memory
L0C
L0C+1
L0C+2
L0C+3
12^ 1^ 101 V
8^ 2 3
k 5 6 1U^
8 9 !T 11.
3J
13 lU 15
The data in the registers is new routed hack one PE and stored
where the first data came from.
PE, PE1
PE2
PE
Register
Memory
L0C
L0C+1
L0C+2
L$C+3
In 10> K 12.
\
3
8
9
13
2
6
5
1J
12'
Ik
11
15
<
The transfers accomplished can he illustrated as shown below:
L0C
L0C+1
L0C+2
L0C+3
PEQ
PE]_
PE2
PE,
11-
The transfers to complete the scrambling would be:
PE PE. PE, PE,
L0C 8 ^2 12
L0C+1 k~* 10 6 Ik
L0C+2 1 9 5 ^11L0C+3
.3 13-" "~T" 15
This final transfer is accomplished by only one route since each
data point moves exactly two PE's away (end around). Once again all PE's
are active. The following diagram shows how this final step is accomplished
PEQ
PE1
PE2
PE3
L0C
L0C+1
8.r 2/ 12
*ir ^LO 6 Ik
L0C+2 i 9 5^ ~±± — -
L0C+3 3, *-13r^ 7 15
1
Thus, after two iterations, the data has been properly ordered
for input to the transform routine.
PEo
PE1
PE2
PE3
L0C 8 k 12
L0C+1 2 10 6 Ik
L0C+2 1 9 5 13
L0C+3 3 11 7 15
-12-
In the operations described above, each PE is always performing
some tasks and the data is never moved to an intermediate memory location
so that the scrambling is done in the minimum number of steps and with 100$
efficiency in the use of PE's. It can be seen that neither of these state-
ments is true about the scheme described at the beginning of this section
or the scheme described by Ackins [3]- Unfortunately this algorithm works2
only on data sets which have N points, where N is the number of PE's in
the array.
Since ILLIAC IV has 6k PE's it can scramble a folded data set
of U096 points 100$ efficiently. The algorithm has been extended to handle
small data sets.
Consider the following example of scrambling a 32 point data set
in an 8 PE ILLIAC. The initial state would be:
L0C
L0C+1
L0C+2
L0C+3
PE, PE. PE, PE. PE, PE, PE, PE.
1 2 V k 5 6 7 V
> 17
10 11
*19 V20 21
Ik
^22
15
'23
2k 25 26 27 28 29 30' 31
K_ \
Now perform the transfers indicated. It should be apparent that
these operations take four rather than two routes since each group of four
PE's cannot be treated separately in an end around fashion. Also these
transfers cannot be 100$ efficient since more than one routing distance is
involved.
The result of the transfers would be:
L0C
L0C+1
L0C+2
L0C+3
PE, PE. PE, PE. PE, PE rPE, PE,
16 ^2 2k k 20 ^r 6 28
8*-^T 10 26 12 <4- ^22 Ik 30
1 17 9 J* 1^ 5 21 13 JT 23
3 25*- "*"ii 27 7 29*- ""15 31
-13-
Now only the above indicated transfers are required to complete
the scrambling process. Note that the process is simply two applications
of the previous method performed in parallel. The final result is:
L0C
L0C+1
L0C+2
L0C+3
PE, PE, PE, PE, PE, PE r PE, PE.
16 8 2k h 20 12 28
2 18 10'
26 6 22 1U 30
1 17 9 25 5 21 13 29
3 19 11 27 7 23 15 31
A generalized method for scrambling data which is folded in
memory can now be given. Data is first scrambled in groups which are com-
posed of as many FE's as there are rows of data by the method developed for2
"square" data sets (referring to N data points in N PE's). Finally, con-
sider these "squares" as units and scramble their position. One further
example will make this last step more clear. Consider 16 data points in an
8 PE array.
First iteration
PEr PE. PE, PE, PE, PE r PE, PE.
L0C
L0C+1
1 2 -3 h 5 6 ^18*-
9 10*"^* D
12 *-^13 Ik** 15
The heavy lines below indicate the second step which completes
the scrambling, giving the proper result.
L0C
L0C+1
PE PE. PE, PE, PE1+
PE, PE6
PE,7
8 2 10 k 12 6 1-4
1 9 3 11 5 13 7 15
•14-
Result
L0C
L0C+1
PE, PE. PE, PE, PE, PE r PE, PE,
8 k 12 2 10 6
...
Ik
1 9 5 13 3 11- 7 15
Although the present implementation allows for only up to U096
data points, larger data sets could be handled and it is easily seen that
scrambling would be 100% efficient, but the core restrictions on the size
of the data set and the size of the program would eventually limit the pro-
gram. A short discussion of these restrictions is included under future
considerations -
Future Considerations
Various extensions to this subroutine have been explored. Some
of them seem impractical while others are quite worthy of further investi-
gation and perhaps implementation. There also has been a suggestion
that a separate subroutine be written to handle several parallel transforms,
each with all of its data within one PE. If all transforms could be done in
groups of 6k this would appear to be a very efficient method; however, if
groups of data are not even multiples of Gk, some PE's will be completely
idle during the whole process. It is this author's opinion that such a
subroutine would be very straightforward to code and would have enough value
to be worth programming at some future time.
The question of expanding the present subroutine to accept data
sets larger than U096 sample is largely decided by available core. There
does not seem to be too much demand for greater capability at this time and
the program itself already consumes approximately 2,500 words of memory,
mostly composed of stored constants. Each increase of a power of two in
the size of data sets transformable will increase the size of the subroutine
by almost a factor of two. Such increases can only be justified by a large
demand for that capability. If and when expansion is warranted, the process
would be straightforward with the possible restrictions of a 6kK limit on
-15-
the size of the data set so that it could be completely core contained.
Problems with formatting and accessing disk files tend to be formidable.
It has been shown by Robinson [h] that two real data sets can be
handled simultaneously as if they composed one complex -data set of the same
size. It was suggested that capability be added for handling real data in
this manner, with hopes of a factor of two increase in efficiency. The
algorithm was studied and appears simple enough but applying it to a
parallel structure like ILLIAC IV, does not seem feasible at this time.
-16-
ILLIAC IV FAST FOURIER TRANSFORM
USERS MANUAL
This report is intended to be a description of how to use the
Fast Fourier Transform subroutine and exactly -what it can do. An explanation
of the Fourier Transform and of how the subroutine works on ILLIAC IV is given
in ILLIAC IV Report #226 entitled "A Fast Fourier Transform Subroutine for
ILLIAC IV. " Also, copies of the assembly language listing of the subroutine are
available from the Center for Advanced Computation and the University of Illinois.
The Fast Fourier Transform is used to transform a discrete time series
of data points into a discrete frequency spectrum. Both input and output data
values are represented as complex numbers in the following way: each number
occupies on 6U-bit word using the 32-bit floating point configuration where the
real part of each number occupies the inner 32 bits. When the input series con-
sists of only real numbers the numbers should be stored only in the outer por-
tion of each word and the inner portions should be set to zero. In any case the
output will always consist of complex numbers and will be stored in the same
locations as the input, thus "overwriting" the input.
Due to the structure of the Fast Fourier Transform algorithm all input
sequences must contain a number of data points which is an even power of two.
The sizes of data sets which can be transformed by the ILLIAC IV subroutine are:
8, 16, 32, 6k, 128, 256, 512, 102^, 20^8, ^096. Other sizes of data sets can
be transformed by "padding" them with zeros on the end until their size is in an
even power of two between 8 and U096 inclusive. The result of such a transform
will be correct with only the frequency spacing of the result being effected.
This subroutine uses the standard subroutine linking conventions as
described in "The ILLIAC IV Software Reference Manual. " When referenced from a
GLYPNIR program, a standard subroutine call statement can be used. When refer-
enced from an assembly language program, the standard call macro can be used.
This macro and an example of its use is listed in Appendix A. The Fast Fourier
Transform subroutine requires three parameters named size, number, and pointer.
These parameters have the following significance.
SIZE
Size gives the number of data points present in each input time
series. Size must be an integer and should be an even power of two between 8
and U096 inclusive. If it is not an even power of two, the next greatest power
-17-
of two will be assumed and will determine the size of the transform and the
number of results returned. Note that the data must still be "padded" with
zeros for the result to be meaningful. If the size specified is zero or greater
than 4096, an error condition will be indicated and no transform will be per-
formed. An error is indicated by filling ACAR2 with all ones.
NUMBER
Number must also be an integer specifying the number of transforms
to be performed during this call to the subroutine. Any number of transforms
can be performed as long as they are all the same size and as long as all of the
input data ia available in one continuous block of memory. If number is set to
zero, no transform will be performed and ACAR2 will be set to all ones.
POINTER
Pointer is a 2^-bit CU address giving the location of the first word
of a continuous block of memory which contains all of the input data for this
call of the transform subroutine. Pointer will also address the first word of
the resulting frequency spectrum(s) when the subroutine is exited. This pointer
is automatically set up by the calling macro which is exhibited in Appendix A.
The actual subroutine is contained in a file called FASTFOURIER/
TRANSFORM which is available on a system tape of the subroutine library. This
file must be properly linked to the calling program. The subroutine is called
by using the entry identifier FFT.
INVERSE TRANSFORM
For performing inverse transforms the only change required is to call
by using the entry identifier FFTINV.
-18-
APPENDIX A
Standard call macro from "The ILLIAC IV Software Reference Manual"
DEFINE CALL &NAME(&PARAMETERS)=
&IF &SIGN(&MFIELD (&NAME)
)
&THEN EXTERNAL &NAME; &FI
BEGIN BLOCK
BEGIN USE 63
LIST: DATA &PARAMETERS;
END;
CLC(2);
SLIT(2) LIST;
END;
clc(3);
SLIT(3) &NAME;
EXCHL(3) $ICR; ##;
Sample calling sequence to execute two I38 point transforms.
CALL FFT(HOWBIG,HOWMANY, WHERE);
WHERE: DATA [128 words of data];
HOWBIG: DATA 6k;
HOWMANY: DATA 2;
19-
APPENDIX B
Demonstration Run
This appendix compares a Fourier Transform performed on a Univac
1108 and the same transform performed on the ILLIAC IV Simulator. The infor-
mation is presented in the form of actual computer listing including the
following: l) A listing of the 256 real input data points. 2) The results of
the Univac run (real part first followed by imaginary part). 3) The main
program used to call the FFT routine for the ILLIAC IV simulation run.
k) The results of the simulation run. Differences in the two results are due
primarily to the difference of the word size used in each case.
-20-
Input Data
o. 0. -5b. -24. 4822. -13007. -9326. 5835.171.67. 1 2 D 4 7 , 3. -696. -1676. 77 34. -2215. -7602.1 21
1
. lo5. -17335. 9888. -12968. 2744. 14245. 10537.11^*1 3. l'WW, 4049. <069. ) 7389. 6574. 18416. 507.|C625. 2 364. -3213. -36 09. -6436. -11208. 3291. 7014.-7/'<0. -3226. -16879. 1217 3. -684 8. -8584V 7979. 13897.12/3. - 1 5 1 B
.
1 1 723. 3/09. 768. -7109. 3905. -17075.17524. -6860. 8704. 1967. -4606. -11202. 16318. 3432.4807. -19886. -6603. -11205. -11151. 762. 7412. 951.31 1>7. 32 70. 11954. 3 3 . -1335. -14578. 17089. "-8457
-6505. -12545. 4537. 5618. -5144. -13695. 1017 0. -3683.-8/o2. -924. -2680. 4/16. -4320. -5671. -11186. 12147.-6bl . 1 1049. -12576. 3387. 1053. -12898. 5476. 1719.
-5 6 92. -9780. 611b. 4889. -15757." ""6186."
-1947. " -1902.9280. 40 03. -13438. 1 34 66. -14 7. -5415. 2584. -13501.4027. -41 76." 223b. -10937. 85 7 3. "-13987." "3345. -1129.
-530?. -9614. -64 15. -1 3410. 1344 7. 1673. -10575. -3613.]o5. -191. 499. 1514. -5063." "19279. "8 32. -5213.
12544. 49 /6. -17782. -H99fa. 10533. -5940. -2264. 7824.6105. -16655.
"
9 24/ 3 7666. 3006. -7317.
"
~ -8 118." 3976."5297. -109 75. 10 253. 16419. 2224. 12068. 392. 1382.~ - 3 1 2 . -238. 1088b. 1551. 2753." 6381 . "8903. -393.
14* ; 23. 11118. -9271. -7248. 1997. -1026. 809. 4387.-3143. 6325. -4 87 7. 195. -496. 17 32. 16910. -3611.-5082. -3918. -5548. -2748. 5597. 3206. -8721. -7531.ill?. 15922. 293b.
"- 4 695." -2051. 18 043. """2013. " -306.
-5539. -901 7, -18840. 10401. 3189. -18287. -3289. -4301.-2925. -3157. 10105. "3154. -7270. -.1 1441. """2041 7" "-374673139, -8121. -12835. 9925. 2178. -6024. -8531. 4158.
j-486] . -8923. 18110. 13781. "3 060/." 51297" 157877" 34287n-1069. -2 Ob 8.
2816.-12438.1816/.
1 2253.1954.
-7699.-279 7.
-10595.-6039.
6591.. -1159.-15/5. 17566. 2517.
-21-
Real Univac 1108 Result
.85580000*04-.23935650*05
. 10475777*06
.40882889*05-.79964560*04
. 16432617*06-.H81901?6*05
. 12018498*06-.928207(12*05-.213009^2*05."3661 06P6.05.601555^6*05. 10175 7^4*06.23466077*05. 126766O5.06
-.50596355*05.11267843*06.576140)0*05.28340147*05.930267^9*05
-.12314756*06.21500096*06
-.195869?4*06.39295759*05.17858835*05.123053h9*06
.30633514+05-.79698125+05-.89308050+05-.11257891+05.11725652+05
.19196831*06-.88314708*05
.12051347+06-.29508836+05
.54568208*05
.50177875*05-.81431963+04-.65641566+05
.62304680*05-.75575159*05.36708912*04.86965552*05
.95429297*04-.94544403+05-. 17295822+06.17787883+06
.54710958*05-.72483458*05
-.28900264+06-.42655679+05
-.66131058*05.11841725*06
-.40119934+05.13246392+06
.41877700*05-.10685043*06
.35511073+05-.34118988+05
.10627192*06
.25315634*05-.61932731+05-.93928612+05
.28277265*05-.86787733*05
-.1 13600 16*06.63162532*05.62923921*05
-.19840547*04-.44793332*05-.35014947*05
. 19538353*06
.6791216R+04-.86787713*05.28277265*05
-"738144223*05-.84544034*05
.67912168*04
.19538353*06
-.22494207*06
.59866334+05-.35014947*05-.44793332*05
.32308614*04
.19907523*06-. 19840547*04.62923921*05
.59887308*05-.25588855*05
.63162532*05-.11360036*06
-.24503150+05.75343836+05
.12305369*06
.17858835*05.25315634+05.106271^2*06
-.11576040+06-.56760156+05
.39295759*05-.19586924*06
-.10685043*06.41877700*05
-.49808327+05-.12489943+06
.21500096*06-.12314756*06
.11841725*06-.66131058*05-.72483458*05.54710958*05
-.35125625+05.55541602+05
.93026729*05
.28340147*05.95541438*05
-.10801400*06.57614010*05.11267843*06
.86965552*05
.36708912*04.18504945*05
-.11136791+06-.50596355*05.12676695*06
-.75575159*05.62304680*05.50177875*05.545682(18*05
-.88314708+05.19196831+06
-.79964560+04.40882889+05.10475777+06
-.23935650+05
-.12558822+00-.12877611+06
.23466077*05
.10175734*06
.60155586*05
.36610626*05-.21300992*05-.92820702*05
.59602295+02
.62851 124+05-.14447113+05.68437002+04
-.12452422+06-.18619060+04
.12018498*06-.88190126*05
.12423822+06-.43274833+05
.12878585+06•.32208377*05•.31424301*05-.14610218*05•.11268548+06.46726594+05.10651084*06-.35035923*04-.1 1844547*06•.22496512*05.23490922*05.24522155+05-.13490052+06-.70621265*05-.87760739*05.63107815*05
_L18619414*05_.fB944333*06^87388329*05.48S61633*05-.35586001*05_-.20365177*05.14108025*06.12"12353b"*06~"
^.42466545*05.11306600*06"^.42466545*05.12123530*06.14108025*06•.20365177*05
L.255R6001*05.48561633*05^87388329*05•."18944333*06^18619414*05.63107815*05^87760739^05•.70621265^05"••13490052*06".24522 155*05^23490922*05•.22496512*05..11844547*06••35035923*04.10651084*06.46726594*05
•.1J268548*L06_••14610218*05L.J 1424 30 1_*05.32208377*05.12878585*06
-.43274833*05.12423822*06-.18619060*04-.12452422*06.68437002*04-.144471 1 3*05.62851 124*05.59602295*02-.12877611*06-. 12558822*06.11136791*06.18504945*05-.10801400*06.95541433*05.55541602*05-.35125625*05-. 12489943*06-.4980832"7*05-.56760156*05-.11576040*06.75343836*05-.24503150*05-.25588855*05".S"988730S*05
.19907523*06
.32308614*04
.5986633^*05-.22494207*06-.84544034*05-.38144223*05-.93928612*05-.61932731*05^.341 18988*05.35511073*05.13246392*06
r.46119934*05-.^2655679*05r. 28900264*06.17787883+06
:. 17295822*06-.94544403*05.95429297*04-.65641566*05-.61431963*04-.29508836*05.12051347*06.11725652*05-.11257891*05-.89308050*05-.79698125*05.30633514*05
.16432637*06
-22-
Imaginary Univac 1108 Result
.00000000
. 12539 7-+9 + 05-,71297144+OS
. 18855b8b+06
.17435297*06
.35799352*05-.626712 31*05-. 1 1224Qb9*06-.92065512*04-.83311300*05-.21239087*05-.78648c>52*04-.11474280*06.10789059*06
-.57622153*05.18087399*05
-.17384096*06-.97109678*05-.38454852*05-.48075970+05_.30471259*05.11034351*06.56991673*05
-.731532/7*05-.14147976*06-.20342460*06.34484995*05
-.75169292+05
.73496373*05~.54843035+05~_ .16088750*05.10316574*06.22981703*06
-.20013107*06-.15650693*06-.50385053+05-.88054139*05.44369014+05
_ .33982734+05-.10571027+06-.12952341*06.13493994+04
_ .72885594+05.88596548*05
-.46812337*05-. 14840 741 +06.12412578+05
-.21966100*05-.299284)9*06.15715059*06
-.94287190+05.32348535*03
-.32348535+03.942H71'<0*05
-. 15715054+06.2992H4 19+06.21966100*05
-. 12412578+05. 1484074] +06.46812337+05
-.8859b54M+05-.72885544+05-. 1 3493994+04.12952341 +06.10571027*06
-.33982734+05-.4^369019+05.880541 19+05.50385053+05.15650693*06.20013107+06
-.22981703+06-.10316574+06-.16088750+05-.54843035*05-.73496373+05.751892'-'2 + 05
-.34484995+05.20342460+06.14147976+06
.73153277+05-.56991673+05-.11034351+06-.30471259+05.48075970+05
~ .38454 852+ 05.97109678+05. 17384046+06
-. 18087399*05.57622153+05
_-, 10789059+06. 1 1474200 + 06^.78648652 + 04.21239087+05.83311300+05.92065512+04.11224059+06.626 71231+05"
-.35799352+05"-.17435297 + 06-.188556«6 + 06^.71297144+05
-.12539749+05
-.15282431+06.34464606+05
-.96626729+05.47544088+05
-.10578316+06-.48ifl9366 + 05.649649/7+05
-. 10760260+06.44764577+05.95072141*05.19846301*06.93203335*05.23333460+05
-.15458620+05.35402116+05. 15852791+06.23991060+05
-.79624333+05-.25320520+05-.14226242 + 06"
^36430140+05.11887141+06
-.74552653+05.188907)8+06
-.92939891+05.12785482+06.33364525+05.19050124+05
. 12846535+06
.44 664971 +04"
-.54591791+05-.74105499+05.48101416+04
" ."19423950 + 05.63503576+05.47605405+04"
-_. 17098704+05-.2843124605
~
-.23472000+06-. 10436177+06_-.44790«95 + 05_."72368868 + 05
-.94768170+05-.45146798+05.21326760+05
-.36626935+05.32980707+05.76239434+05.20518874+05
-.60283592+05"_-,.79 2 34 054 05.
-.27383049+05-.41893763+05.51 172803 + 05.49064532+05.1 1941343 + 05"."4696 7453 + 05-.17344318+06-.13977412+06.47270071 +05.24357188+06.31240915+05.14410096+06-.322141 13+05-.24044316+06-.31654141+06-.22730289+06-.13077045+06". 10901463 + 06-.47217832+05.32640422+05-.12036483+06.405873 19< 05-.11836559+06.12702667+06. 10989458+06-.00000000•. 10989458+06. 12702667 + 06
.11836559+06-"."40587319 + 05.12036483+06
-.32640422+05.47217832+05
-.10901463+06"^13077045+06.22730289+06.31654141+06.24644316+06.322141 13+05
-. 14410096 + 06"
-.31240915+05-.24357188+06-.47270071+05
. 13977412 + 06_. 17344318 + 06-.46967453+05^.11941393 + 05-.49069532+05-^51172803+05.41893783+05.._J_7383049_+05
.79234054*05
.60283592*05-.20518874+05-.76239434+05^32980707 + 05.36626435+05-.21326760+05.45146798+05.947681 70+05-.72368868+05.44790895+05.104361 77*06_.23472000 + 06.28431246+05. 17098704 + 05-.47605405+04-.63503576+05"." 19423950*05-.48101416+04.74105499+05.54591791*05-.44664971+04-. 12846535*06-.19050124+05-.33364525+05-.12785482+06.92939891+05-.18890718+06
.74552653+05-.11887141+06-.36430140+05
. 14226242 + 06
.25320520+05,79624333+05
-.239Q1060+05-. 15852791 + 06-.354021 16+05.15458620+05
-.23333460+05-.93203335+05-. 19846301+06-.95072141+05-.44764577+05.10760260+06
-.64964977+05.48219366+05.10578316+06
-.47544088+05.96626729+05
-.34464606+05..15282431 + 06
-23-
Main Program of ILLIAC IV Simulation
(Data stored in external file referenced by ABC)
BEGIN
EXTERNAL FFT,ABC;
SIZE: DATA 256;
NU: DATA i;
MAIN:
:
SETE E.OR.-E;
SETE1 E.OR.-E;
SLIT(2) ARGLIST;
CLC(3)
SLIT(3) FFT;
EXCHL(3) $ICR;
lit(o) I.ABC+6OO .ABC
DISPLAYR $C0.20:8;
HALT;
ARGLIST: DATA SIZE, NU, ABC;
2k-
Illiac IV Simulation Results
(Outer = Real, Inner = Imaginary)
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-27-
APPENDIX C
Timing Sunmary t
SIZE OFDATA SET
1+096
201+8
102*+
512
256
128
6k
2 x 32
1+ x 16
8x8
Overhead
TIME FORSCRAMBLING
75«0 ju sec
kS.h u sec
53*1 u sec
23.2 \i sec
17.^ /u sec
10.5 u sec
I+.65 ju sec
2.80 in sec
2.1+0 u sec
1.75 jLi sec
TIME FOR'FFT
2233.6 (U sec
IO38. k u sec
1+80.0 ju sec
220.1+ ju sec
100.1+ u Sec
1+5-35 u sec
20.6 ju sec
15.65 ju sec
11.10 in sec
6.75 U- sec
TOTAL
2308.6 n sec
IO96.8 ju sec
533»1 u sec
2I+3.I+ ju sec
II7.8 ju sec
55.85 u sec
25.25 /i sec
18.1+5 u sec
13.50 ju sec
8.50 ju sec|i
13 i± sec
3+x ^
2301+ ju sec
1056 ju sec
1+80 a sec
216 ju sec
96 u sec
1+2 ^ sec
18 ju sec
15 ju sec
12 ju sec
9 u sec
This formula gives a rough estimate of the time required to do the FFTportion where each complex multiply operation takes 3 U sec.
t All times base on 50 n sec clock.
-28-
APPENDIX D
SUBROUTINE LISTINGLISTED nM 0H/1J/71 AM OV » 14
<E'-IN CljMMtNT
THE FAST rnijHIFW Ihansfdhm SUBROUTINE IS COMPOSED OF FOURSL ARATE LUAD MiiOULtS. THE FIRST IS THE MAIM PROGRAM HlCH HANDLESTMr SUBROUTINE |
IN ' h f i F AN|l SFIS UP ThE C'l REGISTERS. ^F -T COMESTMt SCRAMBLING ROUTINE HlCH REORDERS THF lM»Uf nATrt SO TmAT ITrAm .;E TMANSFH 1T|). THEN CUflES THE TmANSfJRM routine whichCU- PLETLS THF TRANSFORM, PERFORMS AM IN'FRSF IF DESEED, ANDrEtURNS rUNTROL TU T HE CALLING UROGRAM. FINALLY ALL UF ThEC,U STANTS USED ri|K ThE TRANrflHM ARE COMPILED IN A SE»ARATF HATAEll E.
THREE HARA»«FIEkS AHE RF-'UIPEO HY THIS PRUliRAM, THEY ARELi?TFU HELO* IN THE riRUF'R Im ''HTCH THEY MUST HE SUPPLIED.
1) ST^F - THF HiimHEH UF HA I A POINTS PER DATA SET.?) NUMUFR " THF NUM-.F.H OF DATA SETS T<J HF TRANSFORMF D .
3) POINTER - ThF CU(? a hTT) AoorFSS OF TmE FIRST LOrATTnN Or DATA,
A USERS MAMijAL rUR THlS ill ROUTINE HAS »EtN PUtL.I$HEO AMI) ISTItLFU ?THE ILLTAC IV FAST FOURJFH TRhNSFURM SU«HUUT I NE USEPSMA- UAL" HY JAMF*; E STEVENS It.
ThF main pur 1 1 on jy ThE pwoc.ram FOLLOWS ANn SETS Up ^uRt^i.sTERS VITH CONIRIIL INFORMATION As SHO*N HE LH j •
LUrAL DATA riUEFrR CL'>M OR AO )
DUf PATTERN)In the scramble sErrhH or Tmf p^nr.RA 1 This REgIstEr ts
UbrU F'lR SAy/ING MODE PATTERNS
1 r F L A G
1
this register is dwioe.j into three i-ifl'>s. tme ftpst. hitnu >er is. is used ro indicate -hETher an inverse transform hasFJtFN RE'JUESTFi), IF -'IT 15 IS SET TU ONE AN INVERSE TRANSFORM wTl.LmE PFKFORMEO, THE SFCOMM FTELO OSES THF NEXT in HITS A m i> Is SIMPL*A rUPY UF THF FjwST PARAMETER Sl'E, WHTCH GIVES THF NUMHFR OF OAT«PllTNTS IN EACH TNHiJI DATA SET. ThE VARIABLE. SI/E, MUST «E AN EVE'PU Ep f)F T«U, TF THE PARAMETER, SI/E. PASSED TCI THF SUBROUTINE ISNUt A PH.iER HE T"iU IT ^ I L L h' ASSUMED THAI TmE NFXT LARgEsT PUwEPOK JO IS UESIRFD A-i) THE PROHf;n VALllE IS THEN GFNEpATEd. ThEFI'AL 2H *ll FKLL) UF THE r'L'A" REGISTER TS LOADED <"JTH THF PFAU'XFSS OF THE rlRSF LJCATiUn IK T AV DATA SET Tl) HE TRANSFORMED.THfS AnnRESS IS E^OAL M THE THTRI) paRAMFIEw. PniNTFR. OlVlDEO BY
b* SO IT IS A PF ADDRESS.LATER* IN THE TRANSFORM SECTION U^ THIS PROGRAM, ThE.
INrllRMATIUN In REGISTER DlULATo IS LOAOFU INTU ACAwO AkjD RFfEREDTU ny THF NAME <U/F.
D*(KFT(IRN)THIS KE^ISTFH IS USED T ' SA>'E ThE RETURN ADDRESS EniiND TN
AC-R1 ^HFN THE SIMRuUTIvE iS ENTERED, THE ACTUAL RFTU»M ALsn USFSAC/>Hl, AN ERROQ KETiIKN 15 FLAG^FU '^Y FILLING ACAR? WITH ALL UNES.
DJ AND D4SAVE AREA niH ACARD AND AC.ARl.
AC f RO( ST 71 . INDFV )
IN THE MAIN SECTION UE THE SOmRdUTInE THIS REGISTER IS L"ADEOWItH ThE VALUE ST^E «. THF FIpSI PARAMFTER). AFTER T hF ST/F TS USED
-29-
TU 1AKE a CH If'! t in Jul .-UA-JCh I -ITS HtfilSTTN IS USED Tfl ^HLD TNOExIN'.I»Nt UHMATIDN i)URT''<i Ink. 5 C rt A -i i- L I •', RPilCESS* AS NOTEn A*n»/F» FOPTur TRANSFORM St C I I N ' Till', pP.MiRAM TuTS RFGlsTtR TS |I)AUF'> FROMRt'.l5TF.R i)1(FLAi;) Aid Pt MIKF.i ! i AmATN PY TMF NAME SI*F,
AC « P1 ( Mi).it )
ThTS Hrr,ISTn< I S NIT MlLFn vITh AMY INITIAL »/ALUF AnU TS USE'THnOljdHflilT TmP ^iiHridijT I-iF f W S'WlNfi MOI)F PATTERNS FHP FNAU|.IN(, p E 5
ac » w?( l hop J
TMTS KF'ilSTf.'K IS '" IT MLLFn /'I T H AMY I N T T I A |_ VALUF Am() TS USETH'j JOl>Hni|I THE S'l^.lijTI ^ Mi'< U riP CONTROL iNriiUMATTliN.
A C ' k i ( c n 1 1 n T )
TMTS PF .TSTf^ CiNTaTNb 4 PufNTFp M THE RU* OF fl A T 4 ML I mG
CU. M Ut'RFH* AS FLL AS i\ PUHIEP Ti) THE LAST PU* C j f- OATa (A L I ^ I T ) ."
IT Is IJSFD TO I'ii)t< p L " 1UPV A'.» M OF. Trp M T NE riHEN THE ToAnjsFuPMTS COMPLFTt. TmTS RFi, IsU'P. ]5 sf T up py THE m ATn SFCTlnN Or ThF
SU 'NlTUTT'lE AS FniLH.SS (1) Tut T'CmFmENT FTFlO IS SFT T n nNF .
(2 Tnf L 1 " 11 r' r L'i TS s 1" T Tn TJ AonWEsS Of T HF FlwSl ph.. nr -iATA
PL S ThE TUfAL l'10-H-.H t'F P' 1 <S S'IPPLTFO (sTpF Sl^F fir T^f nATA SETS1 I ,ts THF NLT1 •*• w UF 1)6 t
(LAST 2 " HITS) t <; s L I T
.SET-. :iT./IOE|) MY 64). ID Tpr AnOoESS FJFL'>Trif i'F APDPFSS Of TMF FTPST HOW 0? nATA:
t X T F R N A
1
-EPt :
fill. 1 '1
.
FFT r FN K Yi » SLT T ( n 1 1)
:
SKIP . 1 s
FFTINV/UNTKYI : Sl.TT(nj l ;
STL(O) h"i;
STL( 1 ) •i;'J;
STL( n '•-) ?. 1
LUAOC2) h C J
:
LOAif * ) *C 1
:
Ai.iT(.n -1 ;
CSHI. ( i ) 1'
LFA.XK J) •
CTSMF I 3 j'> !"> t F m P
c p p r n S 4
! J
SLITC1 ) * H 5
C$IIH( 1 ) %C ^ :
SL.IT(J) = 1 ;
CSHI_(-<) ( l >;
CSHL(O) py
:
CA')O(U) «, c « ••
ALTT(^) 1 ;
LUAi)( <> ) s c .*
;
LOAU J ) sC iJ
L E P < T ( 3 ) » t W P !
CSml( i ) -<5( 1 )
;
A 1 I T ( I ) 1
;
LGAn(? I * C 2
:
C S H R ( 2 1 * »
CSHI. (<>> ^^;CAOO(O) ^C2:S T L f ) ,1)1 s
CSHP( U )? a ;
CAOO( j) sCd:
A(_TT( J) «-l ;
C M H T P ( n ?n ',
SLIT( n = 1 :
I , .i.KNI FUtsE*SK*SEi'SA
'SAASA<FEZFi.
*rl*r,H:
* 4 H
i!F
tFP
«FLMil«pE*h Y
*SH<AU<TN<FL? F- EMF<
•' ! I
MN*FE.1,1
ASH«. a n
*SA'LEASEA rji»
*pil
*SE
P M E X
T FLA1^ til
T FLAvE Ar
VF ACV F k E
tt.m f
TCH F
no ThEATEPAT Pfl
ZFPOASF L
SE. S
V T Nil
C'JNSTSHlF
IFT F
I) si/C^fmETCP S
TTH S
/FPULT iMIl
CPF..ETfH T
V MYIFT F
I) PL) I
V F F L
avf sT f;U|l
E LISSI Till
T TMC
T Thr, FuXT S
d F1
J
A PO,A»l .
T ' 1 P N
iwsrTPST
F SHTHA
WFR. PL
EAOIHTRTMFRlirT
TIN«j
LAG.
E TUNT A
F C ONFCllNPET«Y
MT A
HTRU6 '4 •
I. AT,
HTEKAii P
T^FNT L
S FUN I
N
prMt
iv F L sTATTmFNTs.P Nl) INwFRsE.T ATE^FNT .
R INWEKSF.
ADDMFSS.A P d 11 M E n T A D D P E S S .
A«G"M£NT *ST'/F-*.
ALLEST Pp.;FH OF TWON UP FQilAL TO SIZF.TS IT.T11HN ANl) P L A F, EpRflP.
HnE FLAG .
ACT PIT nr From 63PRUPFP Pn.-R.Sl/F
FLAm HFC, ISTFR.Pr, pnlNTFP.i) APr,i!MF H T AnnRESS.
APr,i|HFMT * N 11 M h E R * .
URN AND FLAG FHPUR.SIZE AN, i fiTV PY 64.P li P n I N T E P .
ARGUMENT *PTTNTEP*.CMau'E PF AOORES?).
RFGlSTfcP.TU rLAli PFGISTER.
FuISTFR.IM AnOK FLO HF kCO.1 M I T .
P LOOP CONTROL.c.pEmfmT rjrLn.i»T T n U n f .
-30-
CROTRC 3) i<>;
SLIT CO »o;
CADDO) SC25SETE F.Un.-t,SETFl E • R • • k- i
C H W S 64?Jump HEHR J
CLCC2)••
compc(? i•
Lntr O) %03;
Lt)L( i ) %\j*\
LOL( 3) *02:EXCHLC 3) * I C -?
'
End FFT.
*PUSIT1HN AODRESS EIFLH.*ZERH OUT ADDRESS F"!£Ln.*set a dress FiELn to pointer.*TIJRN UN
* ALL Pt-S.
*WlJHr> SIZF TO 64,*<iO TO SCRAMHLING RiliiTlNF
*frrhr (no si?E r,TVEi.*SET AGAR ? TO A| L OnES*^RESTORE APARO,RESTORE AcARltSFETCH RFTDRN ADORES*,* wE T
'
ipix To CALLING ^Por,R&H.
-31-
LISTED nNI 08/ 1 J/ / 1 ul * 09 »H
* it
v x
h f:r,
i n
X
<;THA ImL 1 • i SECTION4 ****** X X * X
X X
X *
X X
SIZE
INOE
^ODE
lohp
COUN
*
%
*
%
%
t
%
%
I
Xl
I
I
T:
EXTERNAL
Eqi.
Edll
HFf'CEqii
kEi,c
ECUHEf,C
EOUHEGC
M'TI .PElfvUF.Xl
K S
1
1 1 4 C ') l
n
;
M ,4 1 1 F x s % (1 |
o [ l•» e * C 1
:
i ;
* L i ) ! ) f* s * r 5
irlJU jT »>r a
«
J i
THE PE MF.MnpY L'lCATllHM PH^nlX CONTAINS A SET <1F STX ImoECESFUh FACH PL PACKED MTU flfit «MR<i(SI X BITS PER INOE*. 3A TOTAL).Tnr nEF^t NA M Fp pETr.HlNuE,, „HI*H FOLLOWS mASkS OFF qTTS f«UmpEtnhLx ANU STORES ThE uFSULl I >i REGISTER X AND REGISTER fit ThEPARAMETER TO THf DEFINE UFTERMIUFS "HlCH HITS UF PElNOE* ARF TOHE KTCHEU.
OEFINi FETCHlNOE X
LOASHAHRCLPAShAhLLDSLOX
' STARTSahOwvIANYJPEI ii)FX:
A 4 - n S 1 A K r
'
ff
s a ;
*FETCH PACKED iNnFCEs.*ALIGN TO STARTING MJT POSlTTDN.*7ERii REGISTER A.
JrEcovEr rt T TS OF INTEREST.«]NnFX Til REG S ANU DFr, x.
FKnMANr
STaKIN A
TlnNj [n, H
A A
THrTHr
CU P
I N.-
St q A
OUrU
DEFINc
LACMft ST
hETORTlN(,
LL PF
. RE
LUCAY THAC,U MHONE I
usingOATAMHLIMmLNT
dpEkatAKTlNf,N 1 N G T m
LUC ATT n
S S T "11
1
1
G I S T E R s
T 1 (1 N S R
T THFYL>E Pf G
NPUT Pa
THF nASETS LAG DFAI.T
UF SEPT
I UN i)F ThF ,rKAM,>Ll Nf,
L DC A T 1 Uii IN '" F >E Tfi
r PhF VI IUIS C INTF NTS ilF
M. I N irfHA .j ,L IS AL) E
TAJfcJJUSLY JHfH <1iIPPLTE
X ANU S C tl g T A T t INDFCFbPLf. 1 1 'Fly. i. if thanr«M it T rEHATLn hy sinlb ASbtJ iF Til C. inTaTnRAHE.TEK RE njIWE f
l I S THTA St- T . THIS DFFlNE *
Ri'iErf THA -i 1;'- » A NO IT
Nli .% II H S JUArtt ^ARTTTlFM-FP 1^0. uLL ''FS AH
H'lilTlNE
A ')LSTITMF_ l)F
FTmE fMI) "JlTH
ES KIJH
SFFHS A
PLY PHI)
THF PPMF NUMHFILL DNLCriMPLE
ONS ASE ASSU*
INvOLVNATIONSTINA1 J
ICH ACCTHE FOLS T A R T I m
RL PATTTING ThPER ENAH DF HOY HE USTES THF»ESCRI~EU FNA
Es pnii
IN AMDiin jnomplisluringd AM)ER' J FnF TnDF•<LlNG•tS OFFi) INPtlRTI
EU TN
LEn.
T T n G DATsTuFR pE.TufHFS THTSTNFORMA-
DTSTINA-IN SUCHCFS.PATTERN.STDRAGFScpAh^l-(JM OFTHF FFT
OLOOPi
I N T F R C H A m < , r
BFGTN r^LUC^STL(MUDE)LI Tf LOOP)LIT( INUFX 1
RTLLOSHTLLOX
PATTERN
:
1 »KNiU1/2»l . 1 ;
' 31
t~ s . 1 :
• w;
sx»-l ;
THE SYmmdL DLOOP MUST Rr LOCAL.*SA\/F H(JDF PATTERN.XLUUP COUNTER,XPOUTTNQ TNUEXCLOUP COUNT).<iiESTTNATinN INDFX.tCOMPLETE S TO S TRANSFER.iSlAWTlNG INDEX.^COMPLETE X TD X TRAigSFEP.
-32-
LOR
ldaHTLLnEElLljA
SETESETE1LORSTAHTLLOAKTLCROTR(mOdE)LOEE1LOASETESETFlSTAALITC INDEX)CROTLC MODE)COR(MUnE)TXLTM(LOUP)RTLLOSLnRHTLLOAHTLLoEElLnASETESETFlSTA
ENO
*0(COUNT) \
-4NOH1KiOoEi,:<RI
E.UR.-UE.Oa.-fci«0( C Onwt)i*0(cOUUT) t
0(lMOE*>*SRIrNIMJ*NIJ i( lNnEA) I
«mooe»>bH*
E.UP.-LlE.Or.-Ej*O(0DU U ) I
-1I*NUM( IN0£X)|SPATTERS j
.ULOOPJ* S » 1
1
«0((-nijNT >;
&nu ••/?;
*RJ-&NUMJsmUoE i
- R;
E.O-'.-tiF.n-.-t:»0 ( ("JOINT ) S
> t* :
*FETCH STAJR'JljTE.
XSAVE STAR*FAK£ ENDJENO AROUN
IFILL IN S
HNAHLt AL
*FET CH nES*STORE STAx R0UTF OES*SAVF OEST*FAKE END*FNn AROUN%
*FILL IN*F.NABLE AL
*STQRF OEsKINCRFHFNT^RESTORE M
kmOoIfy MO•RETURN TOXoESTINATI'COMPLETE<FETCH STAXF*CHAN(iF*SAVE.tFAKF LNO*ENn AHOMN<FILL In n
«ENARLE AL
iSTORE E*XC
RTING DATA,
TING DATA,AROUND ROUTE (RIGHT).
BY IROM pES.TARTING DATA.L.
TINATION DATA.RTING DATA in OFST»TINATION DATA rack,INATIUN DATA.AROUND ROUTE (LEFT).
BY UNijM pEs»
ESTINATION - )A TA.L.
T. OATA IN (;TART LO'".
index (- L uop count),ooe pattern (almost).Dt PATTERN,LOCAL DLOnPflNC LOOP>
ON INDFX.S TO S TRANSFER.RT AND OEST. DATA.START AND DEST. DATA.
AROUND ROUTF Or 4NUM/?D BY 4NUM PES.ESTINATION DATA.L.
hAnrEo start And dEsT.
FOR THF SMALLER DAT/. 6FTS (*>'* Akd) S-iALLFR) 'SCRAMBLING CAN HEACtUmRLT ShE{) v ITm UNly HNE ACCFSS TO ONT K<\ m Of sTOpA(;F, Tu|F
INTERCHANGE OPEpAUHN CAN BE UflNE USING ONLY REGISTER TpANSrFRSAN-, RUljTINvi, E/CHAmuE IS A OtFTME «hICM PERFORMS ONE TnTFRCHANgFON UATA FOUND In Kt,ISTFK A, THE DISTANCE SEPARATING ThF TWO wORD.S
TU BE InTERCHANgFD IS T nt UNLY PARAMETER. A proper mudf patternMUcT BE SUPPLIED IN THE CD nlUE REGISTER.
D E F T N r EXCHANGE t Nil i J »
LnR <*A;
RTL -,uu <;
LDEEt ».modf;
lds «.a;
Li>A VR?RTL AS. ->c 1
1 DM}
CSHL(MI)OE) C.NIJMl
LDEF1 htU'jUti
Lda »ii i it
m
<<nATA TO REGISTER R.M<OUTF STARTING DATA,*SELFCT ELEMENTS TU RF EXCHANGE'^PERFORM THE^EXCHANGE .
«ROl|TE RACK TO ThF STArT,^POSITION MUDF PATTERN TO* F ILL IN DESTINATION DATA,
%
HERE t E N T R Y 1 : «
CTSHT(ST/F) <SO»X«;CTSRT(ST/Fi VV#A)6SCTSRT(SIZE) bd.xi^;CTSRKSI/F) Sf.xisn;
33-
Xfil
M6«X 3 2»
X6a«^128X256X512X10?X2oai
x
%
\
i
i
4 I
8t
CTSHT(SIZE)CTSHT(ST^F)CTSrT(SIzE)CTSRT(SIZE>CTSRT<SIZE)CTSBT(SUE)CLC(2)CC1MPC(2)LDL(O)LDL(1
)
LnL( 3)
LXCHLO)JumpJumpj u m p
jumpJumpjumpJumpjumpJump
<>6. < 1 2 C3 »
53*X1024)52#X?04H>Si #440*61;
j
SD J 5
SO 4 I
M)2,T icr?a« J
A 1 6 »
ft 32*u64 :
A128'a2b(S 5
Abl? »
a1 o?a
J
A204HI
tERRHRt DATA SlZT Tun L A *r,E.
*SET ACAR? Tl) ALL ONES.*RESTDkE ACARO.^RESTORE ACArtl.X F LT CM RETURN AonRESs.KRETHRN*
this portion uf th^ pkogkah reorders <scramrles> The data in
surm a way that the results uf the toansfurm routine will come outst°ufntially stored in or mfmury. df'ly one of the follo-mng tenstrhuns uf codf "ill he executed depending on the sl*e of the dataSEtS. THE SCRAMHLlNG PROCESS IS PURELY A DATA TRANSFER PoOrlEMAN<x TS ACCOMPLISHED HJICKLWA FRACTION OF THE TRANSFORM TlMF). THF.pRT MAR Y OptRATInNS ARE MLHUrt v ArcESS A,Nn HOuTlNr,, T hE oThEr IN-
S'dUCTION SET UP INDEXING Ami) L"DP CONTROL.
i409MI FFTCHlNOE* 5f»*SLIT(LUOP) 1»J1.1»LIT(IndEx) * b *l
LUOPIj MTL *SM!LDS *RSRTL M.-l
j
Lox &R!
LOR *0( COUNT)RTL O(LUOP);LDA *0(C r»UNT)STR *0( cDUNT
)
RTL *A»0(I* )F.
STR *0(rOUNT^ALIT( IN >FX) s-1 !
TXLTM(LODP> »ludp1;R T I. i S . 1 J
Los « R
;
LOR i ( n O I » N T )
RTL 12:STR *0< COUNT >
ALIT(COIINT) r*3jTXLTMfC'IIMT ) ,A409h;JUMP FFTl!
A2046 t t FETCHINDEX S 1 J 6 :
LIT(MIJUF) .100 0000lNTERCHAMr,r 32;
ALITCCOUNT) =31!TXLTM(COUriT) »A2f)«a?
J
<0E<L0srU•<dE
*CU*ST*c'J
*FL•<H0
*FE1ST*R0
'' *STXIN*rEadE*co
J *fE*R0
I *ST%\H*r,o
*JUSUE
ooopoooooxoE*IN
ST
l)E
oE>EN
FINE (
np cnuut I nc,
STINATMPLETEARTINGMPl ETETCH STUTF«TCH DEORFUTEORECRF'TURN T
STINATMPLtTETCH ST
DTE ENORF EX
CRF.'ENRACK
MP TDFINE (
OOOOOlFINE (
CRFMENHACK
FETCHNTER.INDEXION I
S TOINDFX TO
ARTINEND A
STINAARTINSTINAST. n
T INDLDO
ION I
S TOART A
D AROCHANGT COUFOR N
TRANSFETCHHI
LARGET COUFOR N
FROM PEINOFX).
(-LUNDFX.S TR
X.
X TR
G OATROUNDTIONG OATTIOMATA I
EX (P.lf IN
NDEX.S TR
NO OEUNO b
ED STNT.EXT DFORMFROM
Dp fUUNT),
ansfEr.
ansfEr.A.
RY 64OATA.A In oFST.data rack.n start loc.-loop cuunt),c loup count)
AnsfEr.ST. DATA.
ART ANO DEST.
ATA SFT.ROUTlNEtPEINOF*).
).
NT.EXT DATA SFT.
-3m-
A 1 2 4 I I
L00P2I
A512I I
L00P3t
A256|
i
L00pa t
Al2«l I
loopsj
jumpFETCHlNOEXLIT(MODE)iNTERCHANr.FLIT(LUOP)LDALIT(MUDE)t-xCHANGE
SETESETElSTAALIT(CI)IJNT)TXlTMCLOnP}TXLT(COUNT)JUMPFETCMINUFXLIT(MUI)E)iNTFRCHAN^rLIT(LOOP)LDALIT(MOl)E)EXCHANGESETESFTF1STAALIT(COUNT)TXLTMCLHUP>TXLT(COUNT)JUMPFETCHlNOEXLITCMODE)INTERCHANGFLIT(LUnP)LOALIT(MOOF)EXTHAN'iFLIT(MQOE)EXCHANGESFTESETElSTAALIT(COUNT)TXLTMCUHJPiTXLT ( CHIINT ^
JUMPFETCHINOEXRTLLOSLOALIT(MUi»F )
EXCHANGESETESETElSTALIT(LOUP)LOALIT(MUDE)E X C H A N r, r
LI T( -iUDDEXCHANGESFTE
FFTU XJUMP TO TRANSFORM ROUTINE.4t>»*> XDEFINE (FETCH FROM PElNnFX).1000004000020 OOOl OOOOO'M16| ^DEFINE (LARr.E).1 » 1 5 • * *LOOP COUNTER.O(CnUNT)! *FETCH DATA.iU/fftoOOUOO'B'
toEriNt..
«ENARLL ALL**
*STORE RESULT.^INCREMENT COUNT.XGU RACK FOR NEXT ROW.*GO RACK FOR NEXT OATA SFT.*JUHP TO TRANSFORM ROUTINE.'UnEFINE (FETCH FROM PETNnFX).
* 10 02 00a 01 00200'! 01 0200 IB I
rti *DEFINE (LARGE).1 »f #01 *LUOP COUNTER.OlCOUNMJ XTETCH OATA.
E.UR."ElE.U'v.-t *
(KCOUNT ) l
*\i
L00P2IXCUUNT. A10?/4 :
FFT1 i
39IOJ
24 J
F.Uw.-LJF.U^.-ElOiCUiNl ) ;
si;.LGllPJJ
$CUUNT» A^l?lFFTl i
33IM
*0EFI,ME."
*ENAR|.t ALL.%
iSTORE RESULT.MNCWFMENT COUNT.*G0 RACK Ff)R NEXT RllW.
*G0 PACK FOR NEXT DATA SFT.<JUMP TO TRANSFUWM RriilTlNE.
'fcuEFINE (FETCH F»nM pElNOF.X).104 21 04^104 21 0«2lO«2lOl«la; XOEFINE (LARGE).t.J.Oi *LUOP COUNTER.0CCHUN1 ) I *FETCH OATA.* 3 h o ( a 1 7 , i J », o } 8 ;
2«J KfiEFTNt.a f f ^ 00 00 Ot> ot 77 a On » 8*
E.U7."LJF.i I .'.-LI
o( coun r i
;
= i;
• LUfiPa ;
ftCUiluT. A/»b6 1
F K T 1 :
?f'.t :
* S . II
*h;*o( rmiNT )
;
SOEFTNE,<F.NAhll ALL.t
* ST ORE RESULT.%\HC.»r ME NT CO'iNT,%r,[) BACK FUR nFaT ROW.*G0 H ACK FUR NEXT UATA SFT.*JUMP Til TRANSFORM »0UTlNF..<i>EFIwE (FETCH FROM pFTNnFX).*KOUTF INDEX.^COMPLLTF S Til
*FETCH OATA.S TRANSFER.
= 5 2S2S2«S2t>? ,S?5? ,3252 e>2t)l«l
1 ; '.OLFHL.F.Hy.-LI *ENARLt ALL,E . U * . • 1 1 *
tO(cOUNT)! *ST0RF RESULT.1 . 1 »o; <t UOP CflUNTER.0(CHUNF ) I *FETCH OATA.s3la(SJl4o3iaiBI30: inEFINE.s?41ff41^n01703770360*Bj1?; SOF.FTNK.
L.U'.-t.s iFiMARLL ALL.
-35-
AMI i
A32» »
A16>>
ABl I
SETE1STA
ALIT(COUNT)TXLTM(LOOP)TxLT(CtHJNT)JUMPLDALlT(MOnE)ExchangeLIT(MOOE)EXCHANGELIT(MUOE)ExchangeSETESETE1STATXLTM(COUNT)JUMPLDALIT(MODE)ExchangeLlT(MOnE)EXCHANGESETESETE1STAtxltm(count)jumpLDALIT(MODE)EXCHANGELIT(MUDE)EXCHANGESETESETE1STA
TXLTM(COUNT)JUMPLDALIT(MODE)EXCHANGESETESETE1STA
TXLTM(COUNT)
E.OR.-0(C°U N
II.L0DP5SCUiJNT
FFT1 I
0(CfJUN•2525231163146M»360004j
E.OR."E.Op.-occoun. A64j
FFUl0(CfiUN•5252515j314006»
E.UR.-e.o*.-0(C0UN.A32>FFT1*0(CUUN252007}
601402>
E.UP.-E 1 s •
"
0(cnnN,A16JTFTl »
0(C 1UN
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I XGO back*JUMP TOXFETCH DA
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1463141*1XqEFINE.
0740003601b'XDEFlNE,XENABLE A
X
*STORE REXGO BACKXJUMP TOXFETCH DA
12525?I8I*o£finl,
0630003141b!xdefinl.xenable a%
«TORE REXf,0 RACKXJUMP TOXFETCH DA
052400252ltt|XOEFINE.
1403006014«8IXnEFlNE.XENABLE A
X
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1202405012181XOEFINE.XENABLE A%
USTORE RE
XGO BACK
SuL T .
T COUNT.FOR OTHER RflW.FOR NEXT DATA |FT.TRANSFORM RQUTlNE*TA.
LL.
SULT.FOH NEXT ()ATA SET.TRANSFORM ROUTINE*TA.
LL.
SULT.FOR NEXT 2 DATA SETS.TRANSFORM ROUTINE.TA.
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SULT.FOR NEXT 4 DATA SETS.TRANSFORM ROUTINE.TA.
LL.
SULT.FOR NEXT » DATA SETS
END HERE.
-36-
ilSTEO nfy { 06/13/71 Ait 09U
% i
% %
% i
sm«
MGDEs
LCOPj
COUNT t
BEGIN* ?. * x x * x
TRANSFORM% % % % %
EXTERNAL
KE6CEQUKrr.c
R?GCEQi.J
REGCTHE FOLLOWING
S 8
SEC ION% %
S I W |
01
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SMlJf)E«*Cl *
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*cnijNT B sriiOEFImE is a
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STANDARD transfer operation used tn
Trtr fU irIEn TRANSFORM ROUTINE. THE dARTTAl RESULTS OF EAcH ITERA-TirN ARE USED In THE PL .mHLRE THEY *ERE CUMPIlTED AS WELL AS IN A
PL SRME OlSTANCF AWAY FnR 1 hL NF'T ITERATION. THE FUNCTION nF THISntp-lNt IS TO MOwE THESE PARTIAL RE SULT S
(
FOUND IN REGISTER A) TO THFpHopER pES uoIST IS A NumhEP TELLlN f, H Ovg F A R). T hF pROpFR hOqEpA T TFRN MUST BE SUPPLIED IN THE CU REGI STER .MODE ( AC A R 1 ) . NnU ThATRL-IsTER MODE (ACAR1) IS NUT CHANGED BY THIS DEFINE,
DEFINr TRANSFERRTLCOMPC(MODE)LDEE1LDSRTLCOMPC(MW>E>L0EE1LDSLDASETESETE1
*DIST»»S A , X D I S T I
I
$ mUoEisr;SA»-4DIST>
t
RMUnE}<a;$rjE.UR.-ElE.Ow,-Ei*»i
*ROUTF REGISTER A RlfiMT.^COMPLIMENT MODE PATTERN.IENARLE PES.<LOAD REGISTER S IN nESlREO PES.*ROUTE RFGISTER A LEFT.^COMPLIMENT MODE PATTFPN.XENARI.E PES.XLOAD REGISTER S IN oFSlRED PES.*LOAD REGISTER A IN SAME PES.^ENABLE ALL PES.
PAGE I
THE following define pehfoms THE COMPLEv multiplicationOPrRATlON ON Two UPERANdS. EACH OPERAND IS ON wORO COMpQsEn OpT*n 32-BIT FLOATING PUlNT NUMBERS* THE OUTER OF wHlCH Is THE REALpApT AND THE INNER IS THE IMAGINARY PART. THE TwO OPERANDS MUSTRE In THE A AND S REGISTERS IN ALL PESt THE RESULT IS |_EfT INRE-.IsTER A. THE RESULTS IN TErvS OR RKREAL PART OF FIRST OPERAND)RiJ.Il.ANn 12 ARF AS FOLLOWS,
R«R1*R2-I 1 *\'i
l«Rl*I2+R2*Il
DEFINr COMPLEXMULTMLRNLORSETECHSASETELOSLOSLDASWAPAM(_RN
LDB
it W H "l I *
%Sl* WHQ I
E.Aij ».-El
E.O..-L:
SA;SH{
I
SRS
SSI
^MULTIPLY REALS AND IMAGINARIES.*WEL0AU SFCOND OPERAND.
*CHANr,E SIGN OF INNEp (I**2*-l).*
triNE OPERAND TO R REGISTER.*SAVE PARTIAL RESuLT.*FIRST OPERAND Tn A REGISTER.XIMAG TO OUTER* REAL To INNER.XMULTlPLY R1*T2 AND R2*Il.^POSITION PARTIAL RESULTS
-37-
IN HEATr m
SHnULG I M E N
MUi Tl
CUmSTTMr p
TlrN.ALl UWTlnNON CO
SWAPXAONTHE FUgIstErEmoryo re e
ITERATHE BARLlCATANTS (
REVIOUTHE
S HALFTO THEMPLEX
OEFINr M
L
C
L
L
A
S
I
S
sT
ANn COCUf'STAwAc TAOF AnGOF ThOFIpST
1
THFM I
OEc InESTnKEOTU BERErUfH
ULTADDAOMPLEOROSONTAOARNTAHE CUsinesNTS R
KENLES I
SE OFOlJADR
HE FUNTO T
ARECONS
THE A
ED IN
LLO*IS An
LOCATQUAlTION.sic nION rW S).S ITrSYMETTHEOTHF
MIJLTI
OSUR
XMULT
I
SB)N<i DEFID PRODUIUNS.TU HALF
THE S
PERATIOETwEEN
THISRATIONRY OF T
OPEHAT1R HALF.PLY UNE
##>NE HEUUICES T ,U
THE OFFSTHE NUMIGNIFICAN OF THEthe resij
new «uAnTU PRUOUHE CALCtj
ONS Tl) R
THUS A
AIJU AND
XFOR FINAL ADDITION,
»ES PRRESULTFT USEPER OFNCE OFFAST
LTS OfTITY I
CE RESLATIONF ELIMSTANDONE S
OPER CONS AND $T
Tn FINROWS BETHIS OE
FOURIERONE ITE
s Then a
ULTS FORS AS «FLINATEO R
ARD OEFIURTRACT
STANTI TO R
OREs T hCm I
TMC FIRITING ORCRATfFINE FOLLOWTRANSFORM I
RATION ANDODEO TO THfTHE PRESEn
L AS THATY ADOINQ ONNE WHICH ACAND PRODUCE
t FnuN ApRORCROn
IN B
1 A C
A SETrEsu
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ndOprI-ANOIN A
RIEF.OMPLE*OF
LTS OFRA-w S
TRAC-ISHESRES*
NSTAnRATh
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ANT S
LLOWTHE PRLEFTTANTsODREsTHE
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&
*,
S
s
$
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(«
THTORET
ES IN
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S UFCURHF
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AN EXBE S
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AND C
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THF M
NT TT
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UR 'JU
FIRSTUSINELUAUSU >UA
h S A
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NT)|«fETch first opErano,NT)»*MULTTPLY BY CONSTANT
XFETCH SECOND QPERANn*SAVE PARTIAL RESULT,*ADD ?ND OPERANO TO PXSTORE ONE RESULT,X?No OPERAND TO RfGlstSUBTRACT PARTIAL RES
NT)»tfl ISTORE THE OTHERTS PROGRAM ARE REpRESENTeTIALS. TO DECREASE THE n
AS PART OF THE PROGRAM a
MO SYMETRY* THE sines orAORANTS ARE SIMPLE TRANSFOUADRANT, FOR THIS REAs
S ARE STORED.A SET OF CONSTANTS ANO T
nRANT VALUES. ThE RESULTND THE MEMURY LOCATIONSRED. THE INPUT PARAMETERLOCATION CONTAINING THE
ON. THE TRANSFORMATION j
(W) IN RQS.
ARTIAL RES.
TER A.
ULT.RESULT,
n As SINESUMBER OFDvANTAGECOSINESormationON ONLY
RANSFORMSS Or ThISF THFTS ASSUMED
CONSTANTSS DEFINED!
KEAL(UUAO ?)»-lMAGINANY(UUAO 1)IMAGINARY(QIIA0 2 ) »PE AL( vJUAn 1)
DEFINr CHANGEUtlAI)LDASWAPASETE1CHSASETE1LOS
<,CUMHLLXNUM1ICCUMPLEXNUMI
;
E#ANO."EI
E.UR.-Ej
XFETCH CONSTANTS.KSWAP HEAL AND IMAGINARY PARTS*%
*CHAMGE SIGN OF REAL PART.I
*LEAVE RESULT IN REGISTER S.
THE FULLOWTNb DEFINE RESLTS THE TRUE-FALSE FLIP-FLOP (TFFF)WHtCH IS USED TO COWTRUL A SMALL LOOP.
OEFINr RESETFF =
CACRB **l XRESET ACR HEfllSTEH -R I T CTFFF)I
* THE FOLLOWING DEFINE SETS ThE TpUE-fAlSE FLIP"FLOP wrICH 15
* USrU TU CONTROL A SMALL LOUo.
OEFINr SETFF a
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THE FOLLOWING COUE IS EXECUTED IF AN INVERSE TRANSFORM ISREQUESTED. THE ONLY OPERATION REQUIRED IS THAT EACH ELEMENT OFTHF SOLUTION M 1
1 s T BE UlVIUEn BY N(SIZE). THE NUMBER OF DATA POINTSIN T M E FUNCTION, SINCE N Is A ROWER OF TWO THIS CAN BE DONE BY
SUBTRACTING LOG(RASE 2)^i FHQH EACH FXPONFNT. THE PROPER EXPONENTSAHr CREATED AT A 1 I HE WHEN NCSIZE) IS KNOW EXACTLY AND THE* AREPUT IN ACAR2CLHHP) JUST HEFORt EXECUTION IS TRANSFEREO TO OUT.
SLITCCOUNT) aQJCADDCCOUNT) SIZE»
XZERO COUNT.XSTART WITH FIRST ROW OF DATA.
-1+2-
INVERSE LOA O(COUNT) J
SBEX SC2ITXLTM(COUNT) .INVERSE!
RETURN! LDL(3) *02|LOL(O) *D3lLOKl) $04>EXCHL(3) SlCRjENO FFT1.
XFETCH DATA.XDIVIDE BY SIZE (NO uNnERFLOW).XDO FOR ALL ROWS OF DATA.* F ET CH RETURN AqdRESs.IRESTORE ACARO.^RESTORE ACAR1,XRETURN TO CALLING PROGRAM.
43-
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FILL l28tSIwUnTRYHiDATA ( PA IRC 1.0* 1, )• PA IR(, 707 10678 12..707 1067
PAIR( 1.0* -1.0). PA I R<-. 70710678 12*. 707 1067812)*PAIR(-1.0»-1.0).PAIR( -.707 10678 12. -.707 1067812),PA I R( -1.0»l,n), P A IRC. 7071067812.-. 707 10 A78t 2 ))8.
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)»PAIR< 0.8314696123).PAIR( 0.5555702330).PAIR{ 0,1950903220.9807852804)*)»PAlR(-0. 5555702330).PAlR(-0. 8314696123),PAlR(-0, 98078528044.-. 195090322).).PAIR("0. 8314696123)*PAlR(-0. 5555702330).PAlR(-0. 1950903220-.9807852804).)»PAIR< 0.5555702330)*PAIr( 0.8314696123AIR( 0.980785?804,-0.0980171403)»).PAIR( 0.9569403357).PAIR( 0.8819212644).PAIR( 0,7730l04 5 34
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)»PAIR("0. 9569403357).PAIR("0. 8819212644),PAlR(-0, 7730104^34). PAIRCO. 6343932842)»PAlP(-0. 4713967368)»PAlR<-0. 2902846773P AIr(-0. 38 ?68 343->4»- 0.9 238795 325
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0.9569403357)0.88l92l2644)0,7730104 534)0.6343932842)0.4713967368)0.2902846773)0,0980171403)
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REFERENCES
[l] Cooley, J. W., Lewis, P. A. W., Welch, P. D., "The Fast Fourier TransformAlgorithm and its Application," IBM Watson Research Center, Yorktown Heights,New York, February, 1967.
[2] Barnes, G. H., Brown, R. M., Kato, M. , Kuck, D. J., Slotnick, D. L., Stokes,R. A., "The ILLIAC IV Computer," IEEE Transactions on Computers C-17, No. 87U6 (1968).
[3] Ackins, G. M., "Fast Fourier Transform via ILLIAC IV," ILLIAC IV Document.
No. 198, July 12, I968.
[U] Robinson, E. A., Multichannel Time Series Analysis with Digital ComputerPrograms, Holden-Day (1967), p. 290.
-6k-
UNCLASSIFIEDSecuritirClaaaification
DOCUMENT CONTROL DATA -RIO(Security elaaalllcallon of till*, body of itiWt) and Indomhtg annotation muat ba xiMwj mhan Ifcg orarall tapott la clatalUad)
I, ORIGIN A TINC AC Tl VI T Y (Corporata author)
Center for Advanced ComputationUniversity of Illinois at Urbana-ChampaignUrbana, Illinois 61801
am. REPORT ICCURITY CLASSIFICATION
UNCLASSIFIED2b. CROUP
3. REPORT TITLE
A FAST FOURIER TRANSFORM SUBROUTINE FOR ILLIAC IV
4. descriptive MOTH (Typa ot rmpmrl and htclualra mwtmm)
Research Report5. AUTHOR(S) (Firat mm, mlddla Initial. Imat nmma)
James E. Stevens, Jr.
8 REPORT DATE
October 15, 1971
fa. TOTAL NO. OF PACE*
67
76. NO. OF REP*
•a. CONTRACT OR 6RANT NO.
DAHC01+-72-C-0001b. PROJECT NO.
ARPA Order 1899
•a. ORieiNATOR** REPORT NUMBCR(S)
CAC Document No. 17
»fc. OTHER REPORT NOlil (Any othar number* that may ba maalfnad(hi* raport)
10. DISTRIBUTION STATEMENT
Copies may be obtained from the address given in (l) above.
II. SUPPLEMENTARY NOTES
None,
12. SPONSORING MILITARY ACTIVITY
U.S. Army Research Office-DurhamDuke^-Station ....
Durham, North jgarolina13. ABSTRACT
This report is a description of a Fast Fourier Transform (FFT)
subroutine written in assembly language for the ILLIAC IV computer. The
subroutine uses the Cooley-Tukey algorithm for performing discrete Fourier
transforms. The parallel nature of the Cooley-Tukey method lends itself
very well to a highly parallel machine like ILLIAC IV. 'Timing simulation
results have shown that this program will perform Fourier transforms
faster than they are being done on any existing computer system.
DD ,'°<,?..1473 UNCLASSIFIEDSecurity Classification
UNCLASSIFIEDSecurity Clasmification
key worn o%
Seismic Data Processing
ASK
Fast Fourier Transform (FFT)
Discrete Fourier Transform (DFT)
Time Series
Frequency Spectrum
UNCLASSIFIEDSecurity Classification
