Time and Frequency Analysis of Discrete-Time Signals

8/12/2019 Time and Frequency Analysis of Discrete-Time Signals

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The function X (e j ) or X ( ) is also called the Discrete-Time FourierTransform( DTFT ) of the discrete-time signal x(n). The inverse #T*T is defined "ythe following integral

Properties of Discrete-Time Fourier Transform concise list of #T*T properties is given in Ta"le .&.

Analog frequency and digital frequencyThe fundamental relation "etween the analog frequency, / , and thedigital frequency, , is given "y the following relation

or alternately,

where T is the sampling period, in sec., and fs = & /T is the samplingfrequency in H0.

1ote, however, the following interesting points

2 The unit of / is radian3sec., whereas the unit of is 4ust radians.2 The analog frequency, / , represents the actual !ysical frequency of t!e "asic analog signal , for example, an audio signal (5 to 6 kH0) or avideo signal (5 to 6 7H0). The digital frequency, , is the transformedfrequency from 8quation . a or 8quation . " and can "e consideredas a mathematical frequency, corresponding to the digital signal.

"a#


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"$#

F%&'( !.1(a) nalog frequency response and (") digital frequency response.

Analog frequency response and digital frequency response9ne of the most important differences "etween discrete-time systems andanalog systems is that discrete-time systems have a periodic frequencyresponse, H (e j ), while analog systems have a nonperiodic *ouriertransform H ( j/) #*igure .& illustrates this difference in "etween H ( j/)and H (e j ).

4.1.) Discrete Fourier Transform

The #iscrete *ourier Transform (#*T) is a practical extension of the#T*T, which is discrete in "oth time and the frequency domains #The#T*T X ( ) is a periodic function with period :; radians. This property isused to the divide the frequency interval (5, :;) into $ points, to yield the#*T of the discrete-time sequence x(n), 5 < n < $ = & as follows

TABL !.)#*T Theorems

The +nverse #iscrete *ourier Transform (+#*T) is given "y the followingequation


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*ro+erties of the DFT concise list of #*T transform properties is given in Ta"le .:. !ome ofthe key features and practical advantages of the #*T are as follows2 The #*T maintains the time sequence x(n) and the frequency sequence

X (% ) as finite vectors having the same length $ . dditionally, as seenfrom 8quation .6 and 8quation . , the #*T and +#*T are "oth finitesums, which makes it very convenient to program these equations oncomputers and microprocessors.2 T!e time-frequency relation is a very important relation in practical#*T applications. The index n corresponds to the time value t = n > t ,sec., where > t is the sampling time interval. The index %correspondsto the frequency value = % > , radians, where > is the #*T outputfrequency interval. Then, for a given $ -point #*T, the time frequencyrelation is given "y

2 The concept of time s!ift in the #*T is defined circularly the sequence x(n) &5 < n< $ = & is represented at $ equally spaced points around acircle as shown in figure .:a, for 1 ?@. Then, a circular shift,represented as x((n = ) @), for example, is implemented "y moving theentire sequence x(n) counter-clockwise "y five points, as illustrated in*igure .:". Hence, the sequence x(n) = A x(5) x(&) x(:) x( ) x(6) x( ) x( )

x(B)C, and the shifted sequence x((n = )@) = A x( ) x(6) x( ) x( ) x(B) x(5) x(&) x(:)C#


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F%&'( !.)(a) !equence x(n) and (") circularly shifted sequence x((n = ) @)#

ircular on olutionn $ -point circular convolution of two sequences x(n) and ! (n) is definedas

y(n) ?

$ote : The sequences x "n#, h"n #, and y"n# ha e the same ector lengthof N .am+le

#etermine the circular convolution of the two @-point discrete-timesequences, x&(n) and x:( n), given "y

SolutionThe @-point circular convolution is given "y


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"a# "$#

F%&'( !.!(a) !equence x(m) and (") reflected sequence x(( 'm )@)#

%ircular convolution can "e carried out either "y analytic tec!niques ,such as the sliding tape method, or "y com uter tec!niques , such as7 TD E. Fe will discuss "oth approaches "elow.The sliding ta e met!od can "e done "y hand calculation, if the num"erof points in the #*T, $& is quite small# The procedure is as follows

2 Frite the sequences x&(m), x:( m) &and x:(( 'm )@) as shown "elow.

The sequence x: ( 'm ) is o"tained from the sequence x: (m) "y writing thefirst element in the vector x: (m), then starting with the last element in

x: (m) and continuing "ac%(ards . Then, the dot product of the vectors x&(m) and x: (( 'm )@) gives the convolution output x(5).!imilarly, the next term in the sequence, x: ((& = m)@), is o"tained "yshifting x&( 'm ) "y one step to the right, and "ac% again to t!e "eginning of t!e )ector . The dot product of the vectors x&(m) and x: ((& ' m )@) givesthe convolution output x(&).

2 lternately, one could arrange the vector elements x&(m) and x:( m) in $ = @ equally spaced points around a circle, as shown in *igure . a. The


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vector x (( 'm ) ) is o"tained "y reflecting the vector elements of x: (m)a"out the hori0ontal axis as shown in *igure . ". The vector x: ((& ' m )@)is o"tained "y shifting the elements of the vector x: (m) "y one positioncounter-clockwise around the circle. Hence, the output vector is x(n) ?

A: : 6 6 :C.Gsing the com uter met!od , the circular convolution of the twosequences, x&(n) and x: (n), can also "e o"tained "y using the convolution

property of the #*T, which is listed as $roperty : in Ta"le .: a"ove.This method consists of three steps.2 Ste+ 1/ 9"tain the @-point #*Ts of the sequences x&(n) and x: (n)

2 Ste+ )/ 7ultiply the two sequences X &(% ) and X :( % )

2 Ste+ !/ 9"tain the @-point +#*T of the sequence X (% ) &to yield thefinal output x(n)

7 TD E program to implement the procedure a"ove is given "elow

$ote* 7 TD E automatically utili0es a radix-: **T if 1 is a power of :.+f 1 is not a power of :, then it reverts to a non-radix-: process.

Fast Fourier TransformThe fast *ourier transform (**T) is an efficient algorithm that is used for

converting a time-domain signal into an equivalent frequency-domainsignal, "ased on the discrete *ourier transform (#*T).


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0.1 % T(2D' T%2The discrete *ourier transform converts a time-domain sequence into anequivalent frequency-domain sequence. The inverse discrete *ouriertransform performs the reverse operation and converts a frequency-

domain sequence into an equivalent time-domain sequence. The fast*ourier transform (**T) is a very efficient algorithm technique "ased onthe discrete *ourier transform, "ut with fewer computations required. The**T is one of the most commonly used operations in digital signal

processing to provide a frequency spectrum analysis A&= C. Two different procedures are introduced to compute an **T the decimation-in-frequency and the decimation-in-time. !everal variants of the **T have

"een used, such as the Finograd transform, the discrete cosine transform(#%T) , and the discrete Hartley transform . $rograms "ased on the

#%T, *HT, and the **T are availa"le in .

3.) D L2*5 T 2F T6 FFT AL&2(%T65 7%T6 (AD%8-)The **T reduces considera"ly the computational requirements of thediscrete *ourier transform (#*T). The #*T of a discrete-time signal

x(nT ) is

where the sampling period T is implied in x(n) and $ is the frame length.The constants + are referred to as twiddle constants or factors, whichrepresent the phase, or

and is a function of the length $ . 8quation ( .&) can "e written for %? 5,&, . . . , $ = &, as

This represents a matrix of $ $ terms, since X (% ) needs to "e calculatedfor $ values of % . !ince ( . ) is an equation in terms of a complexexponential, for each specific %there are approximately $ complexadditions and $ complex multiplications. Hence, the computationalrequirements of the #*T can "e very intensive, especially for large valuesof $ .

The **T algorithm takes advantage of the periodicity andsymmetry of the twiddle constants to reduce the computationalrequirements of the **T. *rom the periodicity of +


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and, from the symmetry of +

*igure .& illustrates the properties of the twiddle constants + for $ ? @.*or example, let %? :, and note that from ( .6), + &5 ? + : , and from ( . ),+ ? = + : .

F%&'( 3.1 $eriodicity and symmetry of twiddle constant + .*or a radix-: ("ase :), the **T decomposes an $ -point #*T into two( $ 3:)-point or smaller #*TIs. 8ach ( $ 3:)-point #*T is furtherdecomposed into two ( $ 36)-point #*TIs, and so on. The lastdecomposition consists of ( $ 3:) two point #*TIs. The smallest transformis determined "y the radix of the **T. *or a radix-: **T, $ must "e a

power or "ase of two, and the smallest transform or the lastdecomposition is the two-point #*T. *or a radix-6, the lastdecomposition is a four-point #*T.

0.! D %5AT%2 -% -F( 9' : FFT AL&2(%T657%T6 (AD%8-)Det a time-domain input sequence x(n) "e separated into two halves


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There are other **T structures that have "een used to illustrate the **T.n alternative flow graph to the one shown in *igure . can "e o"tainedwith ordered output and scram"led input.n eight-point **T is illustrated through an exercise as well as through a

programming example. Fe will see that flow graphs for higher-order**T (larger $ ) can readily "e o"tained.

F%&'( 3.) #ecomposition of $ -point #*T into two ( $ 3:)-point#*TIs, for $ ? @.

F%&'( 3.! #ecomposition of two ( $ 3:)-point #*TIs into four ( $ 36)- point #*TIs, for $ ? @.


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where x,(5), x,(&), . . . , x,(B) represent the intermediate output sequenceafterthe first iteration that "ecomes the input to the second stage.

). At stage )/

The resulting intermediate, second-stage output sequence x,,(5), x,,(&), . . . X ,,(B) "ecomes the input sequence to the third stage.!. At stage !/

Fe now use the notation of X Is to represent the final output sequence.The values X (5), X (&), . . . , X (B) form the scram"led output sequence. and

plot the output magnitude.


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3.4 D %5AT%2 -% -T%5 FFT AL&2(%T65 7%T6 (AD%8-)Fhereas the decimation-in-frequency (#+*) process decomposes anoutput sequence into smaller su"sequences, the decimation-in-time (#+T)is another process that decomposes the input sequence into smaller

su"sequences. Det the input sequence "e decomposed into an evensequence and an odd sequence, or

and x(&), x( ), x( ), . . . , x(: n K &)

Fe can apply ( .&) to these two sequences to o"tainwhich represents two ( $ 3:)-point #*TIs. Det

Then X (% ) in ( .: ) can "e written as

*igure .@ shows the decomposition of an eight-point #*T into two four- point #*TIs with the decimation-in-time procedure. This decompositionor decimation process is repeated so that each four-point #*T is furtherdecomposed into

F%&'( 3.; #ecomposition of eight-point #*T into two four-point#*TIs using #+T


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Time and Frequency Analysis of Discrete-Time Signals