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12: Polyphase Filters
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 1 / 10
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Heavy Lowpass filtering
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 2 / 10
Filter Specification:
Sample Rate:20kHzPassband edge:100Hz (1 = 0.03)Stopband edge:300Hz (2 = 0.09)
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Heavy Lowpass filtering
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 2 / 10
Filter Specification:
Sample Rate:20kHzPassband edge:100Hz (1 = 0.03)Stopband edge:300Hz (2 = 0.09)
Passband ripple: 0.05dB (= 0.006)
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Heavy Lowpass filtering
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 2 / 10
Filter Specification:
Sample Rate:20kHzPassband edge:100Hz (1 = 0.03)Stopband edge:300Hz (2 = 0.09)
Passband ripple: 0.05dB (= 0.006)Stopband Gain: 80dB (= 0.0001)
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Heavy Lowpass filtering
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 2 / 10
Filter Specification:
Sample Rate:20kHzPassband edge:100Hz (1 = 0.03)Stopband edge:300Hz (2 = 0.09)
Passband ripple: 0.05dB (= 0.006)Stopband Gain: 80dB (= 0.0001)
This is an extreme filter because the cutoff frequency is only 1% of the
Nyquist frequency.
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Heavy Lowpass filtering
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 2 / 10
Filter Specification:
Sample Rate:20kHzPassband edge:100Hz (1 = 0.03)Stopband edge:300Hz (2 = 0.09)
Passband ripple: 0.05dB (= 0.006)Stopband Gain: 80dB (= 0.0001)
This is an extreme filter because the cutoff frequency is only 1% of the
Nyquist frequency.
Symmetric FIR Filter:
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Heavy Lowpass filtering
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 2 / 10
Filter Specification:
Sample Rate:20kHzPassband edge:100Hz (1 = 0.03)Stopband edge:300Hz (2 = 0.09)
Passband ripple: 0.05dB (= 0.006)Stopband Gain: 80dB (= 0.0001)
This is an extreme filter because the cutoff frequency is only 1% of the
Nyquist frequency.
Symmetric FIR Filter:
Design with Remez-exchange algorithm
Order = 360
0 1 2 3
-80
-60
-40
-20
0
M=360
(rad/s)
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Heavy Lowpass filtering
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 2 / 10
Filter Specification:
Sample Rate:20kHzPassband edge:100Hz (1 = 0.03)Stopband edge:300Hz (2 = 0.09)
Passband ripple: 0.05dB (= 0.006)Stopband Gain: 80dB (= 0.0001)
This is an extreme filter because the cutoff frequency is only 1% of the
Nyquist frequency.
Symmetric FIR Filter:
Design with Remez-exchange algorithm
Order = 360
0 1 2 3
-80
-60
-40
-20
0
M=360
(rad/s)0 0.05 0.1
-80
-60
-40
-20
0
1
2
(rad/s)
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Maximum Decimation Frequency
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 3 / 10
If a filter passband occupies only a small fraction
of[0, ], we can downsample then upsamplewithout losing information.
0 1 2 3
-60
-40
-20
0
1
2
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Maximum Decimation Frequency
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 3 / 10
If a filter passband occupies only a small fraction
of[0, ], we can downsample then upsamplewithout losing information.
0 1 2 3
-60
-40
-20
0
1
2
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Maximum Decimation Frequency
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 3 / 10
If a filter passband occupies only a small fraction
of[0, ], we can downsample then upsamplewithout losing information.
Downsample:aliased components at offsets of2K
are almost zero because of H(z)
0 1 2 3
-60
-40
-20
0
1
2
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Maximum Decimation Frequency
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 3 / 10
If a filter passband occupies only a small fraction
of[0, ], we can downsample then upsamplewithout losing information.
Downsample:aliased components at offsets of2K
are almost zero because of H(z)Upsample:Images spaced at 2
Kcan be
removed using another low pass filter
0 1 2 3
-60
-40
-20
0
1
2
0 1 2 3
-60
-40
-20
0 = 2/4 K = 4
0 1 2 3
-60
-40
-20
0 = 2/7 K = 7
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Maximum Decimation Frequency
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 3 / 10
If a filter passband occupies only a small fraction
of[0, ], we can downsample then upsamplewithout losing information.
Downsample:aliased components at offsets of2K
are almost zero because of H(z)Upsample:Images spaced at 2
Kcan be
removed using another low pass filter
To avoid aliasingin the passband, we need
2K 2 1 K
21+2
0 1 2 3
-60
-40
-20
0
1
2
0 1 2 3
-60
-40
-20
0 = 2/4 K = 4
0 1 2 3
-60
-40
-20
0 = 2/7 K = 7
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Maximum Decimation Frequency
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 3 / 10
If a filter passband occupies only a small fraction
of[0, ], we can downsample then upsamplewithout losing information.
Downsample:aliased components at offsets of2K
are almost zero because of H(z)Upsample:Images spaced at 2
Kcan be
removed using another low pass filter
To avoid aliasingin the passband, we need
2K 2 1 K
21+2
0 1 2 3
-60
-40
-20
0
1
2
0 1 2 3
-60
-40
-20
0 = 2/4 K = 4
0 1 2 3
-60
-40
-20
0 = 2/7 K = 7
Centre of transition band must be intermediate Nyquist freq, K
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Maximum Decimation Frequency
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 3 / 10
If a filter passband occupies only a small fraction
of[0, ], we can downsample then upsamplewithout losing information.
Downsample:aliased components at offsets of2K
are almost zero because of H(z)Upsample:Images spaced at 2
Kcan be
removed using another low pass filter
To avoid aliasingin the passband, we need
2K 2 1 K
21+2
0 1 2 3
-60
-40
-20
0
1
2
0 1 2 3
-60
-40
-20
0 = 2/4 K = 4
0 1 2 3
-60
-40
-20
0 = 2/7 K = 7
Centre of transition band must be intermediate Nyquist freq, K
We must add alowpass filterto remove the images:
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Maximum Decimation Frequency
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 3 / 10
If a filter passband occupies only a small fraction
of[0, ], we can downsample then upsamplewithout losing information.
Downsample:aliased components at offsets of2K
are almost zero because of H(z)Upsample:Images spaced at 2
Kcan be
removed using another low pass filter
To avoid aliasingin the passband, we need
2K 2 1 K
21+2
0 1 2 3
-60
-40
-20
0
1
2
0 1 2 3
-60
-40
-20
0 = 2/4 K = 4
0 1 2 3
-60
-40
-20
0 = 2/7 K = 7
Centre of transition band must be intermediate Nyquist freq, K
We must add alowpass filterto remove the images:
Passband noise= noise floor at output ofH(z)plus10 log10(K 1) dB.
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1.
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
Example: M= 399,K= 50
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
Example: M= 399,K= 50 R= M+1K
= 8
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
Example: M= 399,K= 50 R= M+1K
= 8
H(z) =M
m=0h[m]zm
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
Example: M= 399,K= 50 R= M+1K
= 8
H(z) =M
m=0h[m]zm
=K1
m=0h[m]zm +
K1m=0h[m+K]z
(m+K) + [Rterms]
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
Example: M= 399,K= 50 R= M+1K
= 8
H(z) =M
m=0h[m]zm
=K1
m=0h[m]zm +
K1m=0h[m+K]z
(m+K) + [Rterms]
=R1
r=0
K1m=0h[m+Kr]z
mKr
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
Example: M= 399,K= 50 R= M+1K
= 8
H(z) =M
m=0h[m]zm
=K1
m=0h[m]zm +
K1m=0h[m+K]z
(m+K) + [Rterms]
=R1
r=0
K1m=0h[m+Kr]z
mKr
=K1
m=0zm
R1r=0 hm[r]z
Kr
wherehm[
r] =
h[m
+K r
]=K1
m=0zmHm
zK
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
Example: M= 399,K= 50 R= M+1K
= 8
H(z) =M
m=0h[m]zm
=K1
m=0h[m]zm +
K1m=0h[m+K]z
(m+K) + [Rterms]
=R1
r=0
K1m=0h[m+Kr]z
mKr
=K1
m=0zm
R1r=0 hm[r]z
Kr
wherehm[
r] =
h[m
+K r
]=K1
m=0zmHm
zK
Example: M= 399,K= 50,R = 8h3[r] = [h[3], h[53], , h[303], h[353]]
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
Example: M= 399,K= 50 R= M+1K
= 8
H(z) =M
m=0h[m]zm
=K1
m=0h[m]zm +
K1m=0h[m+K]z
(m+K) + [Rterms]
=R1
r=0
K1m=0h[m+Kr]z
mKr
=K1
m=0zm
R1r=0 hm[r]z
Kr
wherehm[
r] =
h[m
+K r
]=K1
m=0zmHm
zK
Example: M= 399,K= 50,R = 8h3[r] = [h[3], h[53], , h[303], h[353]]
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Polyphase decomposition
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 4 / 10
For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.
We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).
Example: M= 399,K= 50 R= M+1K
= 8
H(z) =M
m=0h[m]zm
=K1
m=0h[m]zm +
K1m=0h[m+K]z
(m+K) + [Rterms]
=R1
r=0
K1m=0h[m+Kr]z
mKr
=K1
m=0zm
R1r=0 hm[r]z
Kr
wherehm[
r] =
h[m
+K r
]=K1
m=0zmHm
zK
Example: M= 399,K= 50,R = 8h3[r] = [h[3], h[53], , h[303], h[353]]
This is apolyphaseimplementation of the filterH(z)
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Downsampled Polyphase Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 5 / 10
H(z)is low pass so we downsampleits output byKwithout aliasing.
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Downsampled Polyphase Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 5 / 10
H(z)is low pass so we downsampleits output byKwithout aliasing.
The number of multiplications per input
sample isM+ 1 = 400.
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Downsampled Polyphase Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 5 / 10
H(z)is low pass so we downsampleits output byKwithout aliasing.
The number of multiplications per input
sample isM+ 1 = 400.
Using the Noble identities, we can move
the resampling back through the adders
and filters.Hm(zK)turns intoHm(z)
at a lower sample rate.
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Downsampled Polyphase Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 5 / 10
H(z)is low pass so we downsampleits output byKwithout aliasing.
The number of multiplications per input
sample isM+ 1 = 400.
Using the Noble identities, we can move
the resampling back through the adders
and filters.Hm(zK)turns intoHm(z)
at a lower sample rate.
We still perform400multiplications butnow only once for everyKinput
samples.
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Downsampled Polyphase Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 5 / 10
H(z)is low pass so we downsampleits output byKwithout aliasing.
The number of multiplications per input
sample isM+ 1 = 400.
Using the Noble identities, we can move
the resampling back through the adders
and filters.Hm(zK)turns intoHm(z)
at a lower sample rate.
We still perform400multiplications butnow only once for everyKinput
samples.
Multiplications per input sample =8(down by a factor of 50 ) butv[n]has
the wrong sample rate ().
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Polyphase Upsampler
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 6 / 10
To restore sample rate: upsample and
then lowpass filter to remove images
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Polyphase Upsampler
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 6 / 10
To restore sample rate: upsample and
then lowpass filter to remove images
We can use the same lowpass filter,
H(z), in polyphase form:K1
m=0zm
R1r=0 hm[r]z
Kr
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Polyphase Upsampler
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 6 / 10
To restore sample rate: upsample and
then lowpass filter to remove images
We can use the same lowpass filter,
H(z), in polyphase form:K1
m=0zm
R1r=0 hm[r]z
Kr
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Polyphase Upsampler
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 6 / 10
To restore sample rate: upsample and
then lowpass filter to remove images
We can use the same lowpass filter,
H(z), in polyphase form:K1
m=0zm
R1r=0 hm[r]z
Kr
This time we put the delayzm after
the filters.
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Polyphase Upsampler
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 6 / 10
To restore sample rate: upsample and
then lowpass filter to remove images
We can use the same lowpass filter,
H(z), in polyphase form:K1
m=0zm
R1r=0 hm[r]z
Kr
This time we put the delayzm after
the filters.
Multiplications per output sample =400
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Polyphase Upsampler
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 6 / 10
To restore sample rate: upsample and
then lowpass filter to remove images
We can use the same lowpass filter,
H(z), in polyphase form:K1
m=0zm
R1r=0 hm[r]z
Kr
This time we put the delayzm after
the filters.
Multiplications per output sample =400Using the Noble identities, we can move
the resampling forwards through the
filters. Hm(zK)turns intoHm(z)at a
lower sample rate.
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Polyphase Upsampler
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 6 / 10
To restore sample rate: upsample and
then lowpass filter to remove images
We can use the same lowpass filter,
H(z), in polyphase form:K1
m=0zm
R1r=0 hm[r]z
Kr
This time we put the delayzm after
the filters.
Multiplications per output sample =400Using the Noble identities, we can move
the resampling forwards through the
filters. Hm(zK)turns intoHm(z)at a
lower sample rate.
Multiplications per output sample =8(down by a factor of 50 ).
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Complete Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 7 / 10
The overall system implements:
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Complete Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 7 / 10
The overall system implements:
Need an extra gain ofKto compensate for the downsampling energy loss.
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Complete Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 7 / 10
The overall system implements:
Need an extra gain ofKto compensate for the downsampling energy loss.
Filtering at downsampled rate requires16multiplications per input sample(8for each filter). Reduced by K2 from the original400.
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Complete Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 7 / 10
The overall system implements:
Need an extra gain ofKto compensate for the downsampling energy loss.
Filtering at downsampled rate requires16multiplications per input sample(8for each filter). Reduced by K2 from the original400.
H(ej)reaches 10dB at thedownsampler Nyquist frequency of
K.
0 0.05 0.1
-80
-60
-40
-20
0
1
2/50
(rad/s)
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Complete Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 7 / 10
The overall system implements:
Need an extra gain ofKto compensate for the downsampling energy loss.
Filtering at downsampled rate requires16multiplications per input sample(8for each filter). Reduced by K2 from the original400.
H(ej)reaches 10dB at thedownsampler Nyquist frequency of
K.
Spectral components> Kwill be aliased
down in frequency inV(ej).
0 0.05 0.1
-80
-60
-40
-20
0
1
2/50
(rad/s)
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Complete Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 7 / 10
The overall system implements:
Need an extra gain ofKto compensate for the downsampling energy loss.
Filtering at downsampled rate requires16multiplications per input sample(8for each filter). Reduced by K2 from the original400.
H(ej)reaches 10dB at thedownsampler Nyquist frequency of
K.
Spectral components> Kwill be aliased
down in frequency inV(ej).
ForV(ej), passband gain (blue curve)follows the same curve asX(ej).
0 0.05 0.1
-80
-60
-40
-20
0
1
2/50
(rad/s)
0 1 2 3-80
-60
-40
-20
0
1
(downsampled)
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Complete Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 7 / 10
The overall system implements:
Need an extra gain ofKto compensate for the downsampling energy loss.
Filtering at downsampled rate requires16multiplications per input sample(8for each filter). Reduced by K2 from the original400.
H(ej)reaches 10dB at thedownsampler Nyquist frequency of
K.
Spectral components> Kwill be aliased
down in frequency inV(ej).
ForV(ej), passband gain (blue curve)follows the same curve asX(ej).
Noise arises fromKaliased spectralintervals.
0 0.05 0.1
-80
-60
-40
-20
0
1
2/50
(rad/s)
0 1 2 3-80
-60
-40
-20
0
1
(downsampled)
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Complete Filter
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 7 / 10
The overall system implements:
Need an extra gain ofKto compensate for the downsampling energy loss.
Filtering at downsampled rate requires16multiplications per input sample(8for each filter). Reduced by K2 from the original400.
H(ej)reaches 10dB at thedownsampler Nyquist frequency of
K.
Spectral components> Kwill be aliased
down in frequency inV(ej).
ForV(ej), passband gain (blue curve)follows the same curve asX(ej).
Noise arises fromKaliased spectralintervals.
Unit white noise inX(ej)gives passbandnoise floor at 69dB (red curve) even
though stop band ripple is below
83dB(due toK 1aliased stopband copies).
0 0.05 0.1
-80
-60
-40
-20
0
1
2/50
(rad/s)
0 1 2 3-80
-60
-40
-20
0
1
(downsampled)
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Upsampler Implementation
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 8 / 10
We can represent the upsamplercompactly using a commutator.
Sampley[n]comes fromHk(z)wherek =n mod K.
[@f indicates the sample rate]
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Upsampler Implementation
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 8 / 10
We can represent the upsamplercompactly using a commutator.
Sampley[n]comes fromHk(z)wherek =n mod K.
[@f indicates the sample rate]
H0(z)comprises a sequence of7 delays, 7 adders and 8 gains.
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Upsampler Implementation
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 8 / 10
We can represent the upsamplercompactly using a commutator.
Sampley[n]comes fromHk(z)wherek =n mod K.
[@f indicates the sample rate]
H0(z)comprises a sequence of7 delays, 7 adders and 8 gains.
We can share the delays between
all 50 filters.
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Upsampler Implementation
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 8 / 10
We can represent the upsamplercompactly using a commutator.
Sampley[n]comes fromHk(z)wherek =n mod K.
[@f indicates the sample rate]
H0(z)comprises a sequence of7 delays, 7 adders and 8 gains.
We can share the delays between
all 50 filters.
We can also share the gains and
adders between all 50 filters and
use commutators to switch the
coefficients.
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Upsampler Implementation
12: Polyphase Filters
Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 8 / 10
We can represent the upsamplercompactly using a commutator.
Sampley[n]comes fromHk(z)wherek =n mod K.
[@f indicates the sample rate]
H0(z)comprises a sequence of7 delays, 7 adders and 8 gains.
We can share the delays between
all 50 filters.
We can also share the gains and
adders between all 50 filters and
use commutators to switch the
coefficients.
We now need 7 delays, 7 adders and 8 gains for the entire filter.
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Downsampler Implementation
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to formv[i].
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Downsampler Implementation
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to formv[i].
We use the transposed form of
Hm(z)because this will allow usto share components.
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Downsampler Implementation
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to formv[i].
We use the transposed form of
Hm(z)because this will allow usto share components.
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Downsampler Implementation
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to formv[i].
We use the transposed form of
Hm(z)because this will allow usto share components.
We can sum the outputs of the
gain elements using an
accumulatorwhich sums blocks
ofKsamples.
w[i] =K1
r=0 u[Ki r]
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Downsampler Implementation
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to formv[i].
We use the transposed form of
Hm(z)because this will allow usto share components.
We can sum the outputs of the
gain elements using an
accumulatorwhich sums blocks
ofKsamples.
Now we can share all the
components and usecommutators to switch the gain
coefficients.
w[i] =K1
r=0 u[Ki r]
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Downsampler Implementation
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to formv[i].
We use the transposed form of
Hm(z)because this will allow usto share components.
We can sum the outputs of the
gain elements using an
accumulatorwhich sums blocks
ofKsamples.
Now we can share all the
components and usecommutators to switch the gain
coefficients.
We need 7 delays, 7 adders, 8
gains and 8 accumulators in total.
w[i] =K1
r=0 u[Ki r]
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Summary
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 10 / 10
Filtering should be performed at thelowest possible sample rate reduce filter computation byK
actual saving is only K2 because you need a second filter
downsampled Nyquist frequency max(passband) +
2
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Summary
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 10 / 10
Filtering should be performed at thelowest possible sample rate reduce filter computation byK
actual saving is only K2 because you need a second filter
downsampled Nyquist frequency max(passband) +
2
Polyphase decomposition:splitH(z)as
K
1m=0z
mHm(z
K) eachHm(z
K)can operate on subsampled data combine the filtering and down/up sampling
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Summary
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 10 / 10
Filtering should be performed at thelowest possible sample rate reduce filter computation byK
actual saving is only K2 because you need a second filter
downsampled Nyquist frequency max(passband) +
2
Polyphase decomposition:splitH(z)as
K
1m=0z
mHm(z
K) eachHm(z
K)can operate on subsampled data combine the filtering and down/up sampling
Noise floor is higher because it arises fromKspectral intervals thatare aliased together by the downsampling.
S
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Summary
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter
Upsampler
Implementation
Downsampler
Implementation
Summary
DSP and Digital Filters (2015-7295) Polyphase Filters: 12 10 / 10
Filtering should be performed at thelowest possible sample rate reduce filter computation byK
actual saving is only K2 because you need a second filter
downsampled Nyquist frequency max(passband) +
2
Polyphase decomposition:splitH(z)as
K
1m=0z
mHm(z
K) eachHm(z
K)can operate on subsampled data combine the filtering and down/up sampling
Noise floor is higher because it arises fromKspectral intervals thatare aliased together by the downsampling.
Share components between theKfilters
multiplier gain coefficients switch at the original sampling rate
need a new component:accumulator/downsampler(K : )
S
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Summary
12: Polyphase Filters Heavy Lowpass filtering
Maximum Decimation
Frequency
Polyphase decomposition
Downsampled Polyphase
Filter
Polyphase Upsampler
Complete Filter Upsampler
Implementation
Downsampler
Implementation
Summary
Filtering should be performed at thelowest possible sample rate reduce filter computation byK
actual saving is only K2 because you need a second filter
downsampled Nyquist frequency max(passband) +
2
Polyphase decomposition:splitH(z)as
K
1m=0z
mHm(z
K) eachHm(z
K)can operate on subsampled data combine the filtering and down/up sampling
Noise floor is higher because it arises fromKspectral intervals thatare aliased together by the downsampling.
Share components between theKfilters
multiplier gain coefficients switch at the original sampling rate
need a new component:accumulator/downsampler(K : )For further details see Harris 5.