01200 polyasdphase

8/9/2019 01200 Polyasdphase

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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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary


Filter Specification:

Sample Rate: 20 kHzPassband edge: 100 Hz (ω1 = 0.03)Stopband edge: 300 Hz (ω2 = 0.09)


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




Passband ripple: ±0.05 dB (δ = 0.006)


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




Passband ripple: ±0.05 dB (δ = 0.006)Stopband Gain: −80 dB (ǫ = 0.0001)


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary





This is an extreme filter because the cutoff frequency is only 1% of the

Nyquist frequency.


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






Nyquist frequency.

Symmetric FIR Filter:


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






Nyquist frequency.


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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






Nyquist frequency.


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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary


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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




0 1 2 3

-60

-40

-20

0

ω1

ω2

ω


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary





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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






K can be


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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






K can be



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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






K can be



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


We must add a lowpass filter to remove the images:


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






K can be



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


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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary


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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary



Split H (z) into K filters each of order R − 1.


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary



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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




Example: M = 399


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




Example: M = 399 ⇒ R = M +1K

= 8


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




Example: M = 399 ⇒ R = M +1K

= 8

H (z) =M ′

m=0 h[m]z−m


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




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


h0[r] =

h[0] h[50] · · · h[350]

h1[r] =

h[1] h[51] · · · h[351]


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




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


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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary


H (z) is low pass so we downsampleits output by K without aliasing.


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary



The number of multiplications per input

sample is M + 1 = 400.


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary





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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary









We still perform 400 multiplications butnow only once for every K input

samples.


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary









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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary


To restore sample rate: upsample and

then lowpass filter to remove images


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Polyphase Upsampler




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






m=0 z−m

R−1r=0 hm[r]z

−Kr


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Polyphase Upsampler




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






m=0 z−m

R−1r=0 hm[r]z

−Kr


the filters.

Multiplications per output sample = 400


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Polyphase Upsampler




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






m=0 z−m

R−1r=0 hm[r]z

−Kr


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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






m=0 z−m

R−1r=0 hm[r]z

−Kr


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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary


The overall system implements:


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Complete Filter




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary



Need an extra gain of K to compensate for the downsampling energy loss.


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Complete Filter




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary





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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






K .



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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






K .




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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






K .




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




Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary


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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




We can share the delays between

all 50 filters.


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary





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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12: Polyphase Filters• Heavy Lowpass filtering


Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary





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



Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary


We can again use a commutator.The outputs from all 50 filters are

added together to form v[i].


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary




We use the transposed form of

H m(z) because this will allow usto share components.


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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary







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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary






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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary









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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Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary









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



Frequency



Filter


• 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


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Summary



Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary





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



Frequency



Filter


• Complete Filter

• Upsampler

Implementation

• Downsampler

Implementation

• Summary





2


K −

1m=0 z

−mH m(z

K )◦ each H m(z


• Noise floor is higher because it arises from K spectral intervals thatare aliased together by the downsampling.

S


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Summary



Frequency



Filter


• Complete Filter• Upsampler

Implementation

• Downsampler

Implementation

• Summary





2


K −

1m=0 z

−mH m(z

K )◦ each H m(z



• 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



Frequency



Filter


• Complete Filter• Upsampler

Implementation

• Downsampler

Implementation

• Summary




2


K −

1m=0 z

−mH m(z

K )◦ each H m(z



• 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.

01200 polyasdphase

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