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
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 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
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− 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
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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
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Some related techniques:
•Fractional rate conversion
•Multistage upsampling and downsampling
•Polyphase FIR filter
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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?)
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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
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