Polyphase dfee djk u

7/25/2019 Polyphase dfee djk u

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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 (2015-7295) Polyphase Filters: 12 1 / 10


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary


Filter Specification:

Sample Rate:20kHzPassband edge:100Hz (1 = 0.03)Stopband edge:300Hz (2 = 0.09)


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary




Passband ripple: 0.05dB (= 0.006)


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary




Passband ripple: 0.05dB (= 0.006)Stopband Gain: 80dB (= 0.0001)


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






Nyquist frequency.

Symmetric FIR Filter:


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

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

(rad/s)


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

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

(rad/s)0 0.05 0.1

-80

-60

-40

-20

0

1

2

(rad/s)


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Maximum Decimation Frequency



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary


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



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary




0 1 2 3

-60

-40

-20

0

1

2


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary




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



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





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



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






Kcan be


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



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






Kcan be



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



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






Kcan be



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


We must add alowpass filterto remove the images:


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






Kcan be



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


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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary


For our filter: original Nyquist frequency =10kHz and transition bandcentre is at200Hz so we can useK= 50.


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary



We will splitH(z)intoKfilters each of orderR 1.


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary



We will splitH(z)intoKfilters each of orderR 1. For convenience,assumeM+ 1is a multiple ofK(else zero-padh[n]).


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary




Example: M= 399,K= 50


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary




Example: M= 399,K= 50 R= M+1K

= 8


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





= 8

H(z) =M

m=0h[m]zm


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





= 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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





= 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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





= 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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





= 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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





= 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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





= 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


This is apolyphaseimplementation of the filterH(z)


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Downsampled Polyphase Filter



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary


H(z)is low pass so we downsampleits output byKwithout aliasing.


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary



The number of multiplications per input

sample isM+ 1 = 400.


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary









We still perform400multiplications butnow only once for everyKinput

samples.


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary









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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary




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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






m=0zm

R1r=0 hm[r]z

Kr


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Polyphase Upsampler



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






m=0zm

R1r=0 hm[r]z

Kr

This time we put the delayzm after

the filters.


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Polyphase Upsampler



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






m=0zm

R1r=0 hm[r]z

Kr


the filters.

Multiplications per output sample =400


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Polyphase Upsampler



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






m=0zm

R1r=0 hm[r]z

Kr


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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






m=0zm

R1r=0 hm[r]z

Kr


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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary


The overall system implements:


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Complete Filter



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary



Need an extra gain ofKto compensate for the downsampling energy loss.


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Complete Filter



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary




Filtering at downsampled rate requires16multiplications per input sample(8for each filter). Reduced by K2 from the original400.


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Complete Filter



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






K.



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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






K.




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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






K.




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



Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary


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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





H0(z)comprises a sequence of7 delays, 7 adders and 8 gains.


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






We can share the delays between

all 50 filters.


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary







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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary







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



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary


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

added together to formv[i].


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary




We use the transposed form of

Hm(z)because this will allow usto share components.


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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary







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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary






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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary









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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Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary









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


Maximum Decimation

Frequency



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


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Summary


Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





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


Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





2


K

1m=0z

mHm(z

K) eachHm(z


Noise floor is higher because it arises fromKspectral intervals thatare aliased together by the downsampling.

S


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Summary


Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter

Upsampler

Implementation

Downsampler

Implementation

Summary





2


K

1m=0z

mHm(z

K) eachHm(z



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


Maximum Decimation

Frequency



Filter

Polyphase Upsampler

Complete Filter Upsampler

Implementation

Downsampler

Implementation

Summary




2


K

1m=0z

mHm(z

K) eachHm(z



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.

Polyphase dfee djk u

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Transcript of Polyphase dfee djk u