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**Polyphase FIR Filter Implementation for Communication Systems**

DSP C5000 Chapter 20 Polyphase FIR Filter Implementation for Communication Systems

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**Multirate Processing 1 of 2**

Multirate processing arises in many fields of digital signal processing: Digital audio: sampling frequency conversion (32 kHz, 44.1kHz, 48kHz), sharp cut-off of FIR filter, … Signal processing for digital communications: symbol rate processing, bit rate processing, sample rate processing, … Speech processing: 3G speech codec (Adaptive Multi Rate), fractionnal pitch estimation, ... … 3G = Third generation of cellular mobile communications.

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**Multirate Processing 2 of 2**

Involves two actions on the digital signal: Downsampling: resampling downwards the digital signal in the digital domain. Upsampling: resampling upwards the digital signal in the digital domain. M Fe Fe/M Retain one sample over M and discard the M-1 others, every M samples. L Fe LFe Downsampling decreases the sampling rate by discarding samples (decimation). Upsampling adds extra samples (interpolation). Insert L-1 zeros between each sample

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Downsampling 1 of 2 M x(n) y(m) Folding term

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**Downsampling 2 of 2 Anti-aliasing Filter Noble identity for decimation**

x(n) y(m) H(z) Fe Fe/M fc : (Fe/M)/2 The anti-aliasing filter removes frequences above the Nyquist frequency for the new sampling rate. M H(zM) M H(z)

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Upsampling 1 of 2 x(m) y(n) L

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**Upsampling 2 of 2 Interpolating Filter Noble identity for upsampling L**

x(m) y(n) H(z) LFe Fe fC : (Fe/L)/2 The interpolating filter smoothes out the values that were filled with zeroes. Interpolating filter with unity gain have an Equivalent noise bandwidth of 1/L so it cancels 1/Lth of the signal energy. Thus it needs to have a gain of L to preserve the signal magnitude after interpolation. H(zL) L H(z) L

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**Polyphase Implementation of FIR Filters Decimation Case 1 of 4**

H(z) M E(zM) Let n=lM+k with

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**Polyphase Implementation of FIR Filters Decimation Case 2 of 4**

Time Processing load (MAC/s) MTe N M H(z) M E0(zM) E1(zM) EM-1(zM) z-1 Fe Fe/M M-1 filter evaluation over M are discarded. N filter length

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**Polyphase Implementation of FIR Filters Decimation Case 3 of 4**

Using noble identity M E0(z) E1(z) EM-1(z) z-1 Fe Fe/M Time Processing load (MAC/s) MTe N No more useless computations, but one sampling period over M, CPU is burdned with N MAC/s.

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**Polyphase Implementation of FIR Filters Decimation Case 4 of 4**

Equivalent commutator model E0(z) Processing load (MAC/s) E1(z) N/M EM-1(z) MTe Time Fe Fe/M Commutator runs at Fe,. At each input sample only one component is computed and accu- mulated with the result of the previous one. The result is output when the last component is reached and accumulator is reset. This spreads the processing load over MTe.

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**Polyphase Implementation of FIR Filters Interpolation Case 1 of 5**

H(z) R(zL) L Let n=mL+L-1-k with

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**Polyphase Implementation of FIR Filters Interpolation Case 2 of 5**

H(z) Processing load (MAC/s) N R0(zL) R1(zL) RM-1(zL) z-1 L Te/L Time L-1 multiplications by 0 over L For each filter evaluation. N filter length. Fe LFe

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**Polyphase Implementation of FIR Filters Interpolation Case 3 of 5**

Using noble identity R0(z) R1(z) RM-1(z) L At each output sampling instant, only one component is non zero z-1 L L Fe LFe

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**Polyphase Implementation of FIR Filters Interpolation Case 4 of 5**

Equivalent commutator model R0(z) R1(z) RM-1(z) Processing load (MAC/s) N/L Te/L Time Fe LFe For each output sampling instant one polyphase component is computed. When we reach again the first component (M-1) a new input sample is inputed in the delay line of each polyphase component.

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**Polyphase Implementation of FIR Filters Interpolation Case 5 of 5**

Linear Periodically Varying Time system z-1 z-1 hL-1 h2L-1 h3L-1 z-1 h0 h1 hL-1 hL hL+1 h2L-1 h2L+1 h3L-1 h2L z-1 z-1 h1 hL+1 h2L+1 h0 z-1 hL h2L

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**Case Study Shaping filters for a QPSK modem :**

Emitter: interpolation case. Receiver: decimation case Efficient Algorithm Implementation : Good ordering of computations, Efficient memory organization and management.

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**Emitter 1 of 4 QPSK modulator Fb Fs Fe fk**

s(t)=1/2[cos(2pfot).cos(f(Ak,Bk))-sin(2pfot).sin(f(Ak,Bk))] QPSK modulator Cos() RCF DAC Ak fk fk: Phase computation bits Bk Sin() RCF DAC Fb Fs Fe RCF: raised cosine filter DAC: digital to analog converter Bit frequency Symbol frequency Sample frequency

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**Emitter 2 of 4 Let Fe=16Fs (16 sample / symbol)**

Define a raised cosine filter with: 6 symbols length. Roll_off : 0.5 Matlab command h=RCOSFIR(0.5,3,16,1); Equivalent system 16 H(z) In red: ideal interpolating filter In blue: actual RC filter

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**Emitter 3 of 4 The 16 Polyphase filters are defined by :**

Filter length is 97, impulse response is padded with 0 to reach 112=7*16 With N=112 and L=16

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**Emitter 4 of 4 Shuffle coefficients Coefficients Symbols 1st sample**

R=flipud(reshape(h,8,12)); R=round(R*2^15); fid=fopen('coef.inc','wt'); for p=1:8 fprintf(fid,'\t.word\t%d\n,R(p,:)) end fclose(fid); 2nd sample 15th sample When coefficient pointer reaches this address a new symbol will be input at the next output sample period

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**Emitter (C callable) .sect "coefs2"**

Ncomp .set 16 ;number of polyphase component coefs2 .include "coefpoly2.inc" coefsfin: coefsize .set coefsfin-coefs2 Lfil .set coefsize/Ncomp ;polyphase component length filbufQ .usect "filtre2",Lfil ;data buffer .text _firinit: ST #coefs2,*(adbufQ) ;pointer to current coefs pointer STM #filbufQ,AR2 ;zeroed initial buffer condition RPT #Lfil-1 STL A,*AR2+ RET

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**_firTxQ:… ;context save**

LD #var,DP STM #coefsize,BK MVDM adbufQ,AR2 ;current coefs pointer STM #1,AR0 STM #filbufQ,AR3 ;symbol buffer STL A,*AR3 ;new sample (guess hold during 16 samples) RPTZ A,#Lfil-1 ;compute one polyphase component MAC *AR2+0%,*AR3+,A MVMD AR2,adbufQ ;save new current coefs pointer SFTA A,-16 SFTA A,-1 ;output of RCF can be greater than 1 ! ;test if delay symbols is needed BC endTxQ,NTC ;jump if not necessary MAR *+AR3(-2) RPT #Lfil-2 DELAY *AR3- endTxQ: … ;context restore RET

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**Symbol vs Sample Output**

Symbol output Sample output Fe: 16 khz Fs: 1 khz Df : p/4 constant for each symbol f= Fs/8=125 Hz

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**Receiver 1 of 2 Fe Fb Fs Bit processing Symbol processing ADC RCF ADC**

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**Receiver 2 of 2 Receiver structure is quite similar, except that:**

Each polyphase component has its own delay tap Each polyphase output has to be accumulated for M polyphase computations and accumulator is outputed every M input sample and reset. E0(z) E1(z) EM-1(z)

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**Follow on Activities Laboratory 10 for the TMS320C5416 DSK**

Illustrates the effects of decimation and anti-aliasing filters. Laboratory 11 for the TMS320C5416 DSK Illustrates the effects of interpolation and anti-imaging filters. Application 9 for the TMS320C5416 DSK Uses interpolation and decimation to produce sharper cut-offs FIRs than would be obtained otherwise.

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Reference Digital Signal Processing a Practical Approach by Emmanuel C. Ifeachor and Barrie W. Jervis. Chapter 9. Multirate digital signal processing.

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1 Copyright © 2010, Elsevier Inc. All rights Reserved Fig 2.1 Chapter 2.

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