01200 Polyasdphase
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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 (2014-5260) 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 (2014-5260) Polyphase Filters: 12 – 2 / 10
Filter Specification:
Sample Rate: 20 kHzPassband edge: 100 Hz (ω1 = 0.03)Stopband edge: 300 Hz (ω2 = 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 (2014-5260) Polyphase Filters: 12 – 2 / 10
Filter Specification:
Sample Rate: 20 kHzPassband edge: 100 Hz (ω1 = 0.03)Stopband edge: 300 Hz (ω2 = 0.09)
Passband ripple: ±0.05 dB (δ = 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 (2014-5260) Polyphase Filters: 12 – 2 / 10
Filter Specification:
Sample Rate: 20 kHzPassband edge: 100 Hz (ω1 = 0.03)Stopband edge: 300 Hz (ω2 = 0.09)
Passband ripple: ±0.05 dB (δ = 0.006)Stopband Gain: −80 dB (ǫ = 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 (2014-5260) Polyphase Filters: 12 – 2 / 10
Filter Specification:
Sample Rate: 20 kHzPassband edge: 100 Hz (ω1 = 0.03)Stopband edge: 300 Hz (ω2 = 0.09)
Passband ripple: ±0.05 dB (δ = 0.006)Stopband Gain: −80 dB (ǫ = 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 (2014-5260) Polyphase Filters: 12 – 2 / 10
Filter Specification:
Sample Rate: 20 kHzPassband edge: 100 Hz (ω1 = 0.03)Stopband edge: 300 Hz (ω2 = 0.09)
Passband ripple: ±0.05 dB (δ = 0.006)Stopband Gain: −80 dB (ǫ = 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 (2014-5260) Polyphase Filters: 12 – 2 / 10
Filter Specification:
Sample Rate: 20 kHzPassband edge: 100 Hz (ω1 = 0.03)Stopband edge: 300 Hz (ω2 = 0.09)
Passband ripple: ±0.05 dB (δ = 0.006)Stopband Gain: −80 dB (ǫ = 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
ω (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 (2014-5260) Polyphase Filters: 12 – 2 / 10
Filter Specification:
Sample Rate: 20 kHzPassband edge: 100 Hz (ω1 = 0.03)Stopband edge: 300 Hz (ω2 = 0.09)
Passband ripple: ±0.05 dB (δ = 0.006)Stopband Gain: −80 dB (ǫ = 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
ω (rad/s)0 0.05 0.1
-80
-60
-40
-20
0
ω1
ω2
ω (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 (2014-5260) Polyphase Filters: 12 – 3 / 10
If a filter passband occupies only a small fraction
of [0, π], we can downsample then upsamplewithout losing information.
0 1 2 3
-60
-40
-20
0
ω1
ω2
ω
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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 (2014-5260) Polyphase Filters: 12 – 3 / 10
If a filter passband occupies only a small fraction
of [0, π], we can downsample then upsamplewithout losing information.
0 1 2 3
-60
-40
-20
0
ω1
ω2
ω
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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 (2014-5260) Polyphase Filters: 12 – 3 / 10
If a filter passband occupies only a small fraction
of [0, π], we can downsample then upsamplewithout losing information.
Downsample: aliased components at offsets of2πK
are almost zero because of H (z)
0 1 2 3
-60
-40
-20
0
ω1
ω2
ω
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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 (2014-5260) Polyphase Filters: 12 – 3 / 10
If a filter passband occupies only a small fraction
of [0, π], we can downsample then upsamplewithout losing information.
Downsample: aliased components at offsets of2πK
are almost zero because of H (z)Upsample: Images spaced at 2π
K can be
removed using another low pass filter
0 1 2 3
-60
-40
-20
0
ω1
ω2
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /4 K = 4
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /7 K = 7
ω
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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 (2014-5260) Polyphase Filters: 12 – 3 / 10
If a filter passband occupies only a small fraction
of [0, π], we can downsample then upsamplewithout losing information.
Downsample: aliased components at offsets of2πK
are almost zero because of H (z)Upsample: Images spaced at 2π
K can be
removed using another low pass filter
To avoid aliasing in the passband, we need
2πK − ω2 ≥ ω1 ⇒ K ≤
2πω1+ω2
0 1 2 3
-60
-40
-20
0
ω1
ω2
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /4 K = 4
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /7 K = 7
ω
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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 (2014-5260) Polyphase Filters: 12 – 3 / 10
If a filter passband occupies only a small fraction
of [0, π], we can downsample then upsamplewithout losing information.
Downsample: aliased components at offsets of2πK
are almost zero because of H (z)Upsample: Images spaced at 2π
K can be
removed using another low pass filter
To avoid aliasing in the passband, we need
2πK − ω2 ≥ ω1 ⇒ K ≤
2πω1+ω2
0 1 2 3
-60
-40
-20
0
ω1
ω2
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /4 K = 4
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /7 K = 7
ω
Centre of transition band must be ≤ intermediate Nyquist freq, πK
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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 (2014-5260) Polyphase Filters: 12 – 3 / 10
If a filter passband occupies only a small fraction
of [0, π], we can downsample then upsamplewithout losing information.
Downsample: aliased components at offsets of2πK
are almost zero because of H (z)Upsample: Images spaced at 2π
K can be
removed using another low pass filter
To avoid aliasing in the passband, we need
2πK − ω2 ≥ ω1 ⇒ K ≤
2πω1+ω2
0 1 2 3
-60
-40
-20
0
ω1
ω2
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /4 K = 4
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /7 K = 7
ω
Centre of transition band must be ≤ intermediate Nyquist freq, πK
We must add a lowpass filter to 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 (2014-5260) Polyphase Filters: 12 – 3 / 10
If a filter passband occupies only a small fraction
of [0, π], we can downsample then upsamplewithout losing information.
Downsample: aliased components at offsets of2πK
are almost zero because of H (z)Upsample: Images spaced at 2π
K can be
removed using another low pass filter
To avoid aliasing in the passband, we need
2πK − ω2 ≥ ω1 ⇒ K ≤
2πω1+ω2
0 1 2 3
-60
-40
-20
0
ω1
ω2
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /4 K = 4
ω
0 1 2 3
-60
-40
-20
0 ω = 2π /7 K = 7
ω
Centre of transition band must be ≤ intermediate Nyquist freq, πK
We must add a lowpass filter to remove the images:
Passband noise = noise floor at output of H (z) plus 10 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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use 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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[n]).
Example: M = 399
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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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[n]).
Example: M = 399 ⇒ 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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[n]).
Example: M = 399 ⇒ R = M +1K
= 8
H (z) =M ′
m=0 h[m]z−m
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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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[n]).
Example: M = 399 ⇒ R = M +1K
= 8
H (z) =M ′
m=0 h[m]z−m
= K −1
m=0 h[m]z−m +
2K −1
m=K h[m]z−m + · · ·
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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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[n]).
Example: M = 399 ⇒ R = M +1K
= 8
H (z) =M ′
m=0 h[m]z−m
= K −1
m=0 h[m]z−m +
2K −1
m=K h[m]z−m + · · ·
=K −1
m=0
R−1r=0 h[m + Kr]z
−m−Kr
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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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[n]).
Example: M = 399 ⇒ R = M +1K
= 8
H (z) =M ′
m=0 h[m]z−m
= K −1
m=0 h[m]z−m +
2K −1
m=K h[m]z−m + · · ·
=K −1
m=0
R−1r=0 h[m + Kr]z
−m−Kr
=K −1
m=0 z−m
R−1r=0 hm[r]z
−Kr
where hm[r] = h[m + Kr]
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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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[n]).
Example: M = 399 ⇒ R = M +1K
= 8
H (z) =M ′
m=0 h[m]z−m
= K −1
m=0 h[m]z−m +
2K −1
m=K h[m]z−m + · · ·
=K −1
m=0
R−1r=0 h[m + Kr]z
−m−Kr
=K −1
m=0 z−m
R−1r=0 hm[r]z
−Kr
where hm[r] = h[m + Kr]Example:
h0[r] =
h[0] h[50] · · · h[350]
h1[r] =
h[1] h[51] · · · h[351]
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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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[n]).
Example: M = 399 ⇒ R = M +1K
= 8
H (z) =M ′
m=0 h[m]z−m
= K −1
m=0 h[m]z−m +
2K −1
m=K h[m]z−m + · · ·
=K −1
m=0
R−1r=0 h[m + Kr]z
−m−Kr
=K −1
m=0 z−m
R−1r=0 hm[r]z
−Kr
where hm[r] = h[m + Kr]Example:
h0[r] =
h[0] h[50] · · · h[350]
h1[r] =
h[1] h[51] · · · h[351]
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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 (2014-5260) Polyphase Filters: 12 – 4 / 10
For our filter: original Nyquist frequency = 10 kHz and transition bandcentre is at 200 Hz so we can use K = 50.
Split H (z) into K filters each of order R − 1. For convenience, assumeM + 1 is a multiple of K (else zero-pad h[n]).
Example: M = 399 ⇒ R = M +1K
= 8
H (z) =M ′
m=0 h[m]z−m
= K −1
m=0 h[m]z−m +
2K −1
m=K h[m]z−m + · · ·
=K −1
m=0
R−1r=0 h[m + Kr]z
−m−Kr
=K −1
m=0 z−m
R−1r=0 hm[r]z
−Kr
where hm[r] = h[m + Kr]Example:
h0[r] =
h[0] h[50] · · · h[350]
h1[r] =
h[1] h[51] · · · h[351]
This is a polyphase implementation of the filter H (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 (2014-5260) Polyphase Filters: 12 – 5 / 10
H (z) is low pass so we downsampleits output by K without 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 (2014-5260) Polyphase Filters: 12 – 5 / 10
H (z) is low pass so we downsampleits output by K without aliasing.
The number of multiplications per input
sample is M + 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 (2014-5260) Polyphase Filters: 12 – 5 / 10
H (z) is low pass so we downsampleits output by K without aliasing.
The number of multiplications per input
sample is M + 1 = 400.
Using the Noble identities, we can move
the resampling back through the adders
and filters. H m(zK ) turns into H m(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 (2014-5260) Polyphase Filters: 12 – 5 / 10
H (z) is low pass so we downsampleits output by K without aliasing.
The number of multiplications per input
sample is M + 1 = 400.
Using the Noble identities, we can move
the resampling back through the adders
and filters. H m(zK ) turns into H m(z)
at a lower sample rate.
We still perform 400 multiplications butnow only once for every K input
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 (2014-5260) Polyphase Filters: 12 – 5 / 10
H (z) is low pass so we downsampleits output by K without aliasing.
The number of multiplications per input
sample is M + 1 = 400.
Using the Noble identities, we can move
the resampling back through the adders
and filters. H m(zK ) turns into H m(z)
at a lower sample rate.
We still perform 400 multiplications butnow only once for every K input
samples.
Multiplications per input sample = 8 (down by a factor of 50 ) but v[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 (2014-5260) 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 (2014-5260) 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:K −1
m=0 z−m
R−1r=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 (2014-5260) 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:K −1
m=0 z−m
R−1r=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 (2014-5260) 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:K −1
m=0 z−m
R−1r=0 hm[r]z
−Kr
This time we put the delay z−m 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 (2014-5260) 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:K −1
m=0 z−m
R−1r=0 hm[r]z
−Kr
This time we put the delay z−m 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 (2014-5260) 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:K −1
m=0 z−m
R−1r=0 hm[r]z
−Kr
This time we put the delay z−m after
the filters.
Multiplications per output sample = 400Using the Noble identities, we can move
the resampling forwards through the
filters. H m(zK ) turns into H m(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 (2014-5260) 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:K −1
m=0 z−m
R−1r=0 hm[r]z
−Kr
This time we put the delay z−m after
the filters.
Multiplications per output sample = 400Using the Noble identities, we can move
the resampling forwards through the
filters. H m(zK ) turns into H m(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 (2014-5260) 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 (2014-5260) Polyphase Filters: 12 – 7 / 10
The overall system implements:
Need an extra gain of K to 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 (2014-5260) Polyphase Filters: 12 – 7 / 10
The overall system implements:
Need an extra gain of K to compensate for the downsampling energy loss.
Filtering at downsampled rate requires 16 multiplications per input sample(8 for each filter). Reduced by K 2 from the original 400.
-
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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 (2014-5260) Polyphase Filters: 12 – 7 / 10
The overall system implements:
Need an extra gain of K to compensate for the downsampling energy loss.
Filtering at downsampled rate requires 16 multiplications per input sample(8 for each filter). Reduced by K 2 from the original 400.
H (ejω) reaches −10 dB at thedownsampler Nyquist frequency of π
K .
0 0.05 0.1
-80
-60
-40
-20
0
ω1
ω2π /50
ω (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 (2014-5260) Polyphase Filters: 12 – 7 / 10
The overall system implements:
Need an extra gain of K to compensate for the downsampling energy loss.
Filtering at downsampled rate requires 16 multiplications per input sample(8 for each filter). Reduced by K 2 from the original 400.
H (ejω) reaches −10 dB at thedownsampler Nyquist frequency of π
K .
Spectral components > πK will be aliased
down in frequency in V (ejω).
0 0.05 0.1
-80
-60
-40
-20
0
ω1
ω2π /50
ω (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 (2014-5260) Polyphase Filters: 12 – 7 / 10
The overall system implements:
Need an extra gain of K to compensate for the downsampling energy loss.
Filtering at downsampled rate requires 16 multiplications per input sample(8 for each filter). Reduced by K 2 from the original 400.
H (ejω) reaches −10 dB at thedownsampler Nyquist frequency of π
K .
Spectral components > πK will be aliased
down in frequency in V (ejω).
For V (ejω), passband gain (blue curve)follows the same curve as X (ejω).
0 0.05 0.1
-80
-60
-40
-20
0
ω1
ω2π /50
ω (rad/s)
0 1 2 3-80
-60
-40
-20
0
ω1
ω (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 (2014-5260) Polyphase Filters: 12 – 7 / 10
The overall system implements:
Need an extra gain of K to compensate for the downsampling energy loss.
Filtering at downsampled rate requires 16 multiplications per input sample(8 for each filter). Reduced by K 2 from the original 400.
H (ejω) reaches −10 dB at thedownsampler Nyquist frequency of π
K .
Spectral components > πK will be aliased
down in frequency in V (ejω).
For V (ejω), passband gain (blue curve)follows the same curve as X (ejω).
Noise arises from K aliased spectralintervals.
0 0.05 0.1
-80
-60
-40
-20
0
ω1
ω2π /50
ω (rad/s)
0 1 2 3-80
-60
-40
-20
0
ω1
ω (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 (2014-5260) Polyphase Filters: 12 – 7 / 10
The overall system implements:
Need an extra gain of K to compensate for the downsampling energy loss.
Filtering at downsampled rate requires 16 multiplications per input sample(8 for each filter). Reduced by K 2 from the original 400.
H (ejω) reaches −10 dB at thedownsampler Nyquist frequency of π
K .
Spectral components > πK will be aliased
down in frequency in V (ejω).
For V (ejω), passband gain (blue curve)follows the same curve as X (ejω).
Noise arises from K aliased spectralintervals.
Unit white noise in X (ejω) gives passbandnoise floor at −69 dB (red curve) even
though stop band ripple is below−
83 dB(due to K − 1 aliased stopband copies).
0 0.05 0.1
-80
-60
-40
-20
0
ω1
ω2π /50
ω (rad/s)
0 1 2 3-80
-60
-40
-20
0
ω1
ω (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 (2014-5260) Polyphase Filters: 12 – 8 / 10
We can represent the upsamplercompactly using a commutator.
H 0(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 (2014-5260) Polyphase Filters: 12 – 8 / 10
We can represent the upsamplercompactly using a commutator.
H 0(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 (2014-5260) Polyphase Filters: 12 – 8 / 10
We can represent the upsamplercompactly using a commutator.
H 0(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 anduse 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 (2014-5260) Polyphase Filters: 12 – 8 / 10
We can represent the upsamplercompactly using a commutator.
H 0(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 anduse 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 (2014-5260) Polyphase Filters: 12 – 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to form v[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 (2014-5260) Polyphase Filters: 12 – 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to form v[i].
We use the transposed form of
H m(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 (2014-5260) Polyphase Filters: 12 – 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to form v[i].
We use the transposed form of
H m(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 (2014-5260) Polyphase Filters: 12 – 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to form v[i].
We use the transposed form of
H m(z) because this will allow usto share components.
We can sum the outputs of the
gain elements using an
accumulator which sums blocks
of K samples.
w[i] =
K−1
r=0u[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 (2014-5260) Polyphase Filters: 12 – 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to form v[i].
We use the transposed form of
H m(z) because this will allow usto share components.
We can sum the outputs of the
gain elements using an
accumulator which sums blocks
of K samples.
Now we can share all the
components and usecommutators to switch the gain
coefficients.
w[i] =
K−1
r=0u[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 (2014-5260) Polyphase Filters: 12 – 9 / 10
We can again use a commutator.The outputs from all 50 filters are
added together to form v[i].
We use the transposed form of
H m(z) because this will allow usto share components.
We can sum the outputs of the
gain elements using an
accumulator which sums blocks
of K samples.
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] =
K−1
r=0u[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 (2014-5260) Polyphase Filters: 12 – 10 / 10
• Filtering should be performed at the lowest possible sample rate◦ reduce filter computation by K
◦ actual saving is only K 2 because you need a second filter
◦ downsampled Nyquist frequency ≥ max(ωpassband) + ∆ω
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 (2014-5260) Polyphase Filters: 12 – 10 / 10
• Filtering should be performed at the lowest possible sample rate◦ reduce filter computation by K
◦ actual saving is only K 2 because you need a second filter
◦ downsampled Nyquist frequency ≥ max(ωpassband) + ∆ω
2
• Polyphase decomposition: split H (z) as
K −
1m=0 z
−mH m(z
K )◦ each H m(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 (2014-5260) Polyphase Filters: 12 – 10 / 10
• Filtering should be performed at the lowest possible sample rate◦ reduce filter computation by K
◦ actual saving is only K 2 because you need a second filter
◦ downsampled Nyquist frequency ≥ max(ωpassband) + ∆ω
2
• Polyphase decomposition: split H (z) as
K −
1m=0 z
−mH m(z
K )◦ each H m(z
K ) can operate on subsampled data◦ combine the filtering and down/up sampling
• Noise floor is higher because it arises from K spectral 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 (2014-5260) Polyphase Filters: 12 – 10 / 10
• Filtering should be performed at the lowest possible sample rate◦ reduce filter computation by K
◦ actual saving is only K 2 because you need a second filter
◦ downsampled Nyquist frequency ≥ max(ωpassband) + ∆ω
2
• Polyphase decomposition: split H (z) as
K −
1m=0 z
−mH m(z
K )◦ each H m(z
K ) can operate on subsampled data◦ combine the filtering and down/up sampling
• Noise floor is higher because it arises from K spectral intervals thatare aliased together by the downsampling.
• Share components between the K filters
◦ multiplier gain coefficients switch at the original sampling rate
◦ need a new component: accumulator/downsampler (K : Σ)
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 the lowest possible sample rate◦ reduce filter computation by K
◦ actual saving is only K 2 because you need a second filter
◦ downsampled Nyquist frequency ≥ max(ωpassband) + ∆ω
2
• Polyphase decomposition: split H (z) as
K −
1m=0 z
−mH m(z
K )◦ each H m(z
K ) can operate on subsampled data◦ combine the filtering and down/up sampling
• Noise floor is higher because it arises from K spectral intervals thatare aliased together by the downsampling.
• Share components between the K filters
◦ multiplier gain coefficients switch at the original sampling rate
◦ need a new component: accumulator/downsampler (K : Σ)For further details see Harris 5.