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© Gonçalo Tavares, Moisés Piedade1
SPDS – Multirate signal processing I In many applications digital information must be processed at different sampling rates.
Multirate digital signal processing deals with the development of efficient techniques and algorithms for changing the sample rate
Example: in professional audio systems the sampling frequency used is fs = 48kHz but in the compact discs (CD) it is fs = 44.1kHz it may be necessary to convert between these two sampling frequencies
Important applications:1) Spectrum (pulse) shaping in digital modulators2) Signal filtering: if an analog signal is sampled at a rate higher than the minimum
possible value given by the sampling theorem, then the anti-aliasing filter required prior to sampling and the anti-image (reconstruction) filter after D/A conversion may be much more simple
3) Information storage: the information contained in certain signals (for example, voice signals) may be coded and transmitted efficiently using low sampling frequencies. The signal reconstruction is done using multirate signal processing
4) High resolution A/D and D/A conversion: the use of oversampling techniques in A/D and D/A converters allow the quantization noise spectrum to be spread into a bandwidth much larger than the bandwidth required by the signal after filtering the effective number of bits increases (the SNR increases)
5) Narrow bandwidth filtering: the design of narrowband filters (bandwidth much lower than the sampling frequency) is complicated and requires high-order filters (many coefficients). Multirate digital filtering allows these filters to be computed at a much lower sampling frequency and so the filters may be of much lower order
6) Software defined radio (digital radio): multirate techniques allow direct sampling and demodulation of RF signals, without intermediate frequency stages
The fundamental operations in multirate signal processing are decimation and interpolation
© Gonçalo Tavares, Moisés Piedade2
SPDS – Multirate signal processing II Decimation: sampling frequency reduction Sampling frequency reduction by an integer value M that is, fs fs/M In order to avoid aliasing it is necessary to verify the sampling theorem and filter
the signal with an anti-aliasing filter to eliminate signal components with frequency f> fs/(2M). This may lead to information loss
Digital anti-aliasing filter
Decimator(compressor)
Decimation amounts to filtering the original signal and keeping only onesample of the anti-aliasing filter output for every M samples produced. Mathematically w(n) is given by the convolution
and the output y(m) by
( ) ( ) ( ) ( ) ( )k
w n x n h n h k x n k
( ) ( ) ( ) ( )n mMk
y m w n h k x mM k
Example with M = 3
© Gonçalo Tavares, Moisés Piedade3
SPDS – Multirate signal processing III
Decimation: sampling frequency reduction Frequency domain representation: M=3, fs=6kHz
The spectrum of x(n) has components with frequency above fs/(2M) = 1kHz
Frequency response (amplitude) of the Anti-aliasing digital filter
After filtering (the filter is processed at rate fs) the spectrum becomes limited to f < 1kHz
After decimation the spectrum appears repeatedin every multiple of fs/M
© Gonçalo Tavares, Moisés Piedade4
SPDS – Multirate signal processing IV
Interpolation: sampling frequency increase Sampling frequency increase by an integer factor L that is, fs Lfs To keep the information on the Nyquist band which is now [0 Lfs/2], it is necessary to
use an anti-image digital filter to eliminate the signal components with frequency in the interval [fs/2 Lfs/2]
Anti-imagedigital filterInterpolator
(expander)
Interpolation amounts to inserting L-1 zero-valued samples by each sample of the original signal and then filter the result. Mathematically, w(n) is given by
and the filter output by the convolution
, 0, 2 ,( )
0, other
mx m L L
w n Lm
( ) ( ) ( ) ( ) ( )k
y m w m h m h k w m k
Example with L = 3
© Gonçalo Tavares, Moisés Piedade5
SPDS – Multirate signal processing V
Interpolation: sampling frequency increase Frequency domain representation: L = 3, fs = 2kHz
The spectrum of x(n) has repetitions at every multiple of fs = 2kHz
After interpolation to Lfs = 6kHz, the initial image around fs is now in the Nyquist frequency band
Frequency response (amplitude) of the anti-image digital filter
After filtering the original spectrum is obtained but now with a sampling frequency Lfs = 6kHz
© Gonçalo Tavares, Moisés Piedade6
0x
0
1x2x
3x
0x3x
0x
1x2x
3x
2 sT 3 sT
3
3
20 0 sE x T
3 sT0 sT
sT
20 0 sE x T
20 0 3
sTE x
sT3sT0
snT
snT
snT2 sT 3 sT
( )sx nT
( )D sx nT
( )I sx nT
SPDS – Multirate signal processing VI
Sampling frequency conversion may affect the signal energy
Example: Computation of the energy in an interval t = Ts
Decimation keeps the signal energy on average
Interpolation reduces the signal energy by a factor L necessary to compensate this effect by multiplying the samples by L
12
0
1lim ( )
K
x s sK
n
E x nT TK
1 2
0
1lim
3D
Ks
x s xK
n
nTE x T E
K
12
0
1lim
3 3I
Ks x
x sK
n
T EE x nT
K
Interpolation may be viewed as the digital equivalent of an D/A conversion in which the analog signal is recovered by sample interpolation (effect of the anti-image or reconstruction filter)
Decimation may lead to information loss
Interpolation replicates information in frequency bands of interest
© Gonçalo Tavares, Moisés Piedade7
SPDS – Multirate signal processing VII
Sampling frequency conversion by a non-integer, rational factor
Example: When converting a CD to DAT it is necessary to interpolate samples by a factor 48/44.1, which is not integer but is rational
The factor is represented by the rational number L/M (in the CDDAT example: 48/44.1=160/147) and the conversion corresponds to a decimation with a factor L followed by a decimation with a factor M
To avoid possible information loss, the interpolation should be done before the decimation
Example with L = 3 and M = 2
These two filters are operated at the same sampling frequency and
may thus be combined
© Gonçalo Tavares, Moisés Piedade8
SPDS – Multirate signal processing VIII Frequency domain analysis: L=3,
M=2, fs=2kHz
Multistage conversion: when large sampling frequency variations are required (either decimation or interpolation) it is better to use multiple stages because then, less selective (lower order) filters may be used
Example: decimator with M=16
032sf sf
2sf
08sf sf
2sf
16M
1 2 4M M
Difficult
Easier
© Gonçalo Tavares, Moisés Piedade9
SPDS – Multirate signal processing IX
Interpolators: Polyphase filter structure
Exploit the fact that at the interpolator output, in every L samples, L-1 are zero and do not contribute to the filter output do not need to be computed
Example: interpolator with L=3 and FIR anti-image filter with N=9 coefficients
Each sample x(n) originates 3 samples y(m), each of each computed as the output of one of possible 3 FIR sub-filters (L sub-filters in general) with 3 coefficients (N/L in general)
Each sub-filter operates at the lowest, pre-interpolation sampling frequency, fs
Economy: N N/L multiplications, additions, memory
(0)x
(0)x
(0)x
(0)x
(0)x
(0)x
(0)x
(0)x
(0)x
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
(1)x
(0)w
(1)x
(1)x
(1)x
(1)x
(1)x
0
0
0
0
0
0 0
0 0
(2)x
(2)x
(2)x
0
0 0
(8)w
(0), (3), (6)w w w
(3)w (5)w(1)w (2)w (4)w (6)w (7)w
(1), (4), (7)w w w
(2), (5), (8)w w w
2
3
{ (0), (3), (6)}
{ (1), (4), (7)}
{ (2), (5), (8)}
h h h
h h h
h h h
1s
s
s
© Gonçalo Tavares, Moisés Piedade10
SPDS – Multirate signal processing X The interpolator is equivalent to:
In each phase, the interpolator output is taken as the corresponding sub-filter output
The polyphase filter coefficients are:
Delay line withN/L samples
0,1, , - 1
( ) ( ), 0,1, , 1
k
k L
n h k nL Nn
L
Bank of Lpolyphase FIR filters
Structure used in practice
© Gonçalo Tavares, Moisés Piedade11
Decimators: Polyphase filter structure
The filters operate with the lowest, pos-decimation
sampling frequency (fs/M)
Economy: N N/M multiplications, additions
The polyphase filter coefficients are:
SPDS – Multirate signal processing XI
0 1 8
0 3 6
1 4 7
2 5 8
(0) (1) (8)
(0) (3) (6)
(1) (4) (7) output of 3 filters
(2) (5) (8)
x h x h x h
x h x h x h
x h x h x h
x h x h x h
Structure used in practice
Bank of Mpolyphase FIR filters
0,1, , - 1
( ) ( ), 0,1, , 1
k
k M
n h k nM Nn
M
© Gonçalo Tavares, Moisés Piedade12
SPDS – Multirate signal processing XII Applications1) Spectrum (pulse) shaping in digital modulators
bits
( )R sx nT
Complex symbol attribution
( )sg nT
( )I sx nT( )sg nT
( )ss nT
02 cos snT
02 sin snT
Pulse shaping filters
F
Modulator
Constellation
G f
1
T
0 5.
00 1
2
T
1
2T
1
2
Tf
1
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-5 -4 -3 -2 -1 0 1 2 3 4 5
Tempo normalizado, t /T
Raised cosine
pulses with excess bandwidth
/(2T ) Hertz
=0 35.
=0 0.
=10.
© Gonçalo Tavares, Moisés Piedade13
Impulse response of the continuous time filter: raised cosine filter
FIR filter with N coefficients obtained by truncating the impulse response:
With L = 4 samples per symbol:
The interpolator output may be computed with a polyphase structure (4 filters)
SPDS – Multirate signal processing XIII
2sin / cos /
( )/ 1 4 /
t T t Tg t
t T t T
1
-1
1
1 0 0 0 -1 0 0 0 1 0 0 0 1
g0 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13
0
( )R sx nT
Sample delay line
Coefficients
1
1
0
( ) ( ) ( ), 0,1, , 1
NL
Ri
y k nL g k iL x n iL k L
Only these matter
|( ) ( ) 0, , 1ss t nTg nT g t n N
© Gonçalo Tavares, Moisés Piedade14
2) Audio compact disc
To avoid saturating the power amplifier (and intermodulation distortion) it is necessary to filter spectrum images around every multiple of the sampling frequency fs=44.1kHz. This requires very good (and expensive…) analog filters because the frequency band between sucessive replicas is very small
By digital interpolating the signal by a factor L=4 the frequency band between replicas increases and the analog filter requirement is less stringent (and cheaper…)
SPDS – Multirate signal processing XIV
0 20 [kHz]f44.1sf
4LEasier
88.2
0 20 [kHz]f
44.1 88.2 132.3 4 176.4sf
Very difficult
© Gonçalo Tavares, Moisés Piedade15
SPDS – Multirate signal processing XV
New processingcontext
0 [kHz]f
interpolation L M
0 [kHz]f
Easierdecimation M
Difficult to realize
0 [kHz]f2sf sf
sf
2sfM
sfM
Results in the desired filter
sf
2sfM
sfM
3) Narrow-band filtering In a narrow-band filter the passband is very small when compared to the sampling
frequency. The resulting FIR filter has many coefficients which is undesirable because it:
1. Increases the numerical error susceptibility (due to finite precision arithmetic)
2. Increases the computational effort
3. Increases the memory requirement
The effect of these problems may be reduced using multirate signal processing:
Lowpass filtering
Passband filtering
© Gonçalo Tavares, Moisés Piedade16
4) Oversampling (sigma-delta) A/D converters
Quantization noise: quantizer with quantization intervals
Quantizer with B bit:
Admitting that the quantization error e is uniformly distributed over the interval [-/2, /2], its mean value is zero and the variance is, by definition
When converting a sinusoid with amplitude A, the signal power is S=A2/2 and the signal-to-quantization noise is, in dB
SPDS – Multirate signal processing XVI
A 0 A Amplitude2 2
0 e
1( )p e
2 2/22 2 2 21
/2( ) 2
12 3BA
p d d
2 22 1 2B B
A A
2
2 2 2
/2SQNR 10 log 10 log 1.76 6 [dB]
2 /3B
S AB
A
© Gonçalo Tavares, Moisés Piedade17
SPDS – Multirate signal processing XVII
The useful frequency band is the interval [-fs/2, fs/2] with width fs. The power spectral density of the noise is assumed constant in this band and its value is
If the signal of interest occupies only a smaller frequency band [-fmax, fmax] then we may use a lowpass filter in this band and reject the noise in the rest of the band. This may reduce the noise power significantly
2
( ) , 2 2s s
s
f fS f f
f
22( )
max
max
f max
f s
fP S f df
f
2sf
2sf0
maxfmaxf
2
( )s
S ff
Useful band
f
Inverse of the sampling factor
Princípio: ao aumentar a frequência de amostragem para um valor acima do necessário para representar o sinal (=2fmax), o ruído de quantificação é espalhadonuma banda de frequência de largura e a sua potência pode ser reduzida do factor de sobreamostragem
22( )
max
max
f max
f s
fP S f df
f
Principle: By using a sampling frequency higher than that required to represent the signal (=2fmax), the quantization noise is spread in a frequency band of width fs and its power may be reduced by a factor equal to the oversampling factor
2s
max
fF
f
© Gonçalo Tavares, Moisés Piedade18
SPDS – Multirate signal processing XVIII
Example: Consider an audio signal with unilateral bandwidth 10kHz. Determine the sampling frequency to be used by a one bit converter such that it is equivalent to a 6 bit conversion
The minimum sampling frequency with a 6 bit converter is fs=2fmax=20 kHz and the power of the quantization noise is
To have the same noise power with a one bit converter we should have
and so:
2 122 2(6 bit)
3
A
2 2 22 10
2 12
(1 bit) 2 / 3(6 bit) 2
2 / 3
AF
F A
20kHz 20.48MHzsf F
It is possible to obtain additional noise power reduction by combining oversampling with noise shaping techniques
© Gonçalo Tavares, Moisés Piedade19
SPDS – Multirate signal processing XIX Consumer electronic applications, particular audio applications require high
performance, cheap A/D and D/A converters These specifications are difficult to attain with usual conversion techniques like
successive approximations, flash, dual slope, etc… due to conversion errors like Sample & Hold errors, linearity, monotonicity, etc…
One bit converters: do not use S/H, have less errors and are cheap. Performance is attained by:
Oversampling Noise shaping Decimation techniques to obtain digital multibit words
Model:
Due to negative feedback the system will have at the input a signal with zero mean (the one bit A/D is simply a comparator with very high gain)
In this way the output signal is a sequence of bits in which the density of ones represents the amplitude of the input signal
© Gonçalo Tavares, Moisés Piedade20
The signal processing model is:
Transfer function (first order filter):
The signal x(n) passes to the output without modification but the quantization noise is affected by the transfer function N(z), which is highpass
The noise in the signal band is heavily attenuated
SPDS – Multirate signal processing XX
The quantization noise is added here (comparator)
11
1
( )
1( ) ( ) ( ) ( )
1 ( ) 1 ( )
N z
Y z E z X z z Y zz
X z z E z
© Gonçalo Tavares, Moisés Piedade21
By each octave (sampling frequency duplication) the increase in SNR due to oversampling is 3dB. The remaining increase is due to the noise-shaping filter: 6dB for a first order filter, 12dB for second order, 18dB for third order. In general, the SNR increase is (3+6n)dB for a nth order filter
SPDS – Multirate signal processing XXIDigital filter amplitude response. This filter is the decimator anti-aliasing filter and may be implemented with polyphase structures. By rejecting most of the quantization noise spectrum the effective number of bits is increased (the SNR increases)
Signal spectrum after filtering, before decimation. The decimation does not affect the SNR.
The anti-aliasing filter introduces a group delay (which is constant because the filter is a FIR with linear phase). This is a significant inconvenient because there will be a large delay between the analog signal and the corresponding output digital sample
© Gonçalo Tavares, Moisés Piedade22
5) Software-defined radio (digital radio) Example: Wideband base station digital radio (Watkins-Johnson Inc.)
The entire RF band of interest, with an width equal to 14MHz is converted into the baseband by a single demodulator
The signal is then converted by the A/D at a rate fs=30.72 MHz, slightly greater than 214MHz=28MHz. This real signal has all the information channels
Each channel is recovered digitally (software) by a complex quadrature demodulator which operates at a rate fs
After demodulation the (complex) signal is decimated by a factor M=384 and becomes sampled at a rate fs2=80kHz
SPDS – Multirate signal processing XXII
2sf
15.36MHz2sf0
f
2sf
15.36MHz2sf0
f
0( )H fsffN
2j nNep
© Gonçalo Tavares, Moisés Piedade23
Decimators and interpolators for software-defined radio: CIC (Cascade Integrator Comb) filters
Efficient computing structures which may be used in any application requiring large decimation/interpolation ratios
Cascade of N integrators HI(z) with N comb filters with transfer function C(z)
SPDS – Multirate signal processing XXIII
Integrator Comb filter
11
1( ) 11 ( )
1( ) 1
NMI
M
H z zz H zzC z z
sin2( )
sin2
Ns
s
M T
H j Tzeros at , 1, , 1s
k
ff k k M
M
© Gonçalo Tavares, Moisés Piedade24
SPDS – Multirate signal processing XXIV The frequency bands which may be
responsible for aliasing (after decimation) are strongly attenuated
CIC filter disadvantages:
1) The DC gain is |H(0)|=MN which may assume very large values and thus require memory registers with many bits
2) It is difficult to control the filter characteristics: only the number of sections N may be programmed (other versions of these filters allow for another parameter in the comb section which controls the location of the zeros and thus increases filter versatility
CIC filter advantages:
1) Does not require multiplications
2) Does not require coefficients to be stored in memory (coefficients are either -1 or +1)
3) The filter structure is regular, easy to replicate
Note: Two’s complement arithmetic should be used. In this case intermediate overflows or underflows will not influence the system output (provided the output is representable within the dynamic range of the arithmetic)