# Multirate

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Published on February 7, 2008

Author: aiQUANT

Source: slideshare.net

Multirate Digital Signal Processing Basic rate-changing components: upsampler and downsampler: time domain and frequency-domain models 1

Upsampler: increases the sampling rate by an integer factor L Synonyms: rate expander; expander; oversampler x[n] L xU [n]  x[n / L] n = 0, ± L, ±2 L,... xU [n] =   0 otherwise 2

 x[n / L] n = 0, ± L, ±2 L,... xU [n] =   0 otherwise Upsampling keeps the original samples and introduces L − 1 zero samples between them: x[n] t xU [n] t L=7 3

 x[n / L] n = 0, ± L, ±2 L,... xU [n] =   0 otherwise Upsampling keeps the original samples and introduces L − 1 zero samples between them: x[n] T t xU [n] T′ t T′ = T / L f s′ = Lf s 4

Downsampler: decreases the sampling rate by an integer factor M Synonyms: rate compressor; compressor; undersampler; decimator x[n] M xD [ n ] xD [n] = x[nM ] 5

xD [n] = x[nM ] downsampling keeps the 0th, Mth, 2Mth … original samples and skips the rest: x[n] t xD [ n ] t M =7 6

xD [n] = x[nM ] downsampling keeps the 0th, Mth, 2Mth … original samples and skips the rest: T x[n] t xD [ n ] T′ t T ′ = MT f s′ = f s / M 7

Time- and frequency-domain models  x[n / L] n = 0, ± L, ±2 L,... Upsampler xU [n] =   0 otherwise % % X U ( z ) = X ( z L ) : X U ( f ) = X ( Lf ) Action:Shrinking of the frequency axis by a factor L 8

Time- and frequency-domain models Downsampler xD [n] = x[nM ] M −1 1 X D ( z) = M ∑ k =0 k X ( z1/ M ωM ) : X D ( f ) = ? % Action: complicated 9

Upsampler (incorporating LP Postfilter): increases the sampling rate by an integer factor L Synonyms: rate expander; expander; oversampler; interpolator xU [n] x[n] L LPF xI [ n ] fs f s′ = Lf s f s′ = Lf s  x[n / L] n = 0, ± L, ±2 L,... xU [n] =   0 otherwise n xI [n] = h ∗ xU [n] = ∑ h[n − m]x[m / L] m =0 10 assuming both h and x are causal

Upsampling keeps the original samples and interpolates L − 1 zero samples between them, then lowpass filters the result to remove spectral images: x[n] t xU [n] t xI [ n ] t L=7 11

X(f ) L=2 f − fs − fs / 2 0 fs / 2 fs XU ( f ) Images f − fs − fs / 2 0 fs / 2 fs 12

X(f ) L=2 f − fs − fs / 2 0 fs / 2 fs XU ( f ) Anti-imaging Filter Images f − fs − fs / 2 0 fs / 2 fs XI ( f ) Filtered Images f 13 − fs − fs / 2 0 fs / 2 fs

Downsampler (incorporating LP Prefilter): decreases the sampling rate by an integer factor M Synonyms: rate compressor; compressor; undersampler; decimator xL [ n ] x[n] LPF M xD [ n ] fs fs f s′ = f s / M xL [n] = h ∗ x[n] nM xD [n] = xL [nM ] = ∑ h[nM − m]x[m] m =0 14 assuming both h and x are causal

Downsampling lowpass filters to the OUTPUT half-Nyquist bandwidth, then keeps the 0th, Mth, 2Mth … original samples and skips the rest: x[n] t xL [ n ] t xD [ n ] t 15 M =7

Without lowpass prefiltering aliasing occurs: M =2 X(f ) f − fs − fs / 2 0 fs / 2 fs XD( f ) X ( f / 2 + fs ) X ( f / 2 − fs ) Overlap Overlap f − fs − fs / 2 0 fs / 2 fs Aliasing 16

With lowpass prefiltering aliasing is prevented: M =2 XL( f ) f − fs − fs / 2 0 fs / 2 fs XD( f ) X L ( f / 2 + fs ) X L ( f / 2 − fs ) f − fs − fs / 2 0 fs / 2 fs No Aliasing 17

Some related techniques: •Fractional rate conversion •Multistage upsampling and downsampling •Polyphase FIR filter 18

Fractional rate conversion: R = L/M fs f s′ = Lf s f s′ = Lf s x[n] L LPF xU [n] h1 ∗ xI [ n ] h2 ∗ xIL [n] xI [ n ] LPF M xR [ n ] f s′ = Lf s f s′ = Lf s f s′′ = Lf s / M Now combine the two LPFs 19

Fractional rate conversion: R = L/M fs f s′ = Lf s f s′′ = Lf s / M x[n] L LPF M xR [ n ] h∗ h[ n] = h1 ∗ h2 [n] NB: L and M must be relatively prime, having no common factor (why?) 20

Polyphase FIR filter Example: 11th-order FIR filter, requiring 12 (6 different) coefficients H ( z ) = h[0] + h[1]z −1 + h[2]z −2 + h[3]z −3 + h[4]z −4 + h[5]z −5 + h[5]z −6 + h[4]z −7 + h[3]z −8 + h[2]z −9 + h[1]z −10 + h[0]z −11 H ( z ) = E0 ( z 3 ) + z −1 E1 ( z 3 ) + z −2 E2 ( z 3 ) * where E0 ( z ) = h[0] + h[3]z −1 + h[5]z −2 + h[2]z −3 E1 ( z ) = h[1] + h[4]z −1 + h[4]z −2 + h[1]z −3 E2 ( z ) = h[2] + h[5]z −1 + h[3]z −2 + h[0]z −3 21

x[n] 3 E0 ( z ) + y[n] z −1 3 E1 ( z ) + z −1 3 E2 ( z ) Each of the 3 3rd-order FIR filters requires 4 coefficients, but they all work at the reduced rate, and this is advantageous; e.g. reduced power consumption 22

Question: Apply the symmetry-exploitation trick to the polyphase filter, and redraw the block diagram 23

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