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DSP: Multirate Filter Banks

Digital Signal ProcessingMultirate Filter Banks

D. Richard Brown III

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DSP: Multirate Filter Banks

Multirate Filter Banks and Applications

Example of a two-channel analysis and synthesis multirate filter bank:

For example: H0 is a LPF and H1 is HPF.

Example applications:

Oversampling in digital audio systems. Subband coding of speech and image signals. Encryption/security.

See Vaidyanathan, P.P., Multirate digital filters, filter banks, polyphasenetworks, and applications: a tutorial, Proceedings of the IEEE , vol.78,

no.1, pp.5693, Jan 1990.D. Richard Brown III 2 / 8

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DSP: Multirate Filter Banks

Conditions for Perfect Reconstruction

In the absence of any processing between the down/up-samplers, underwhat conditions on H0, H1, G0, and G1 will we have y[n] =x[nM]?

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DSP: Multirate Filter Banks

Conditions for Perfect Reconstruction

In the absence of any processing between the down/up-samplers, underwhat conditions on H0, H1, G0, and G1 will we have y[n] =x[nM]?

Lets see what happens when H0=G0 are ideal lowpass filters andH1=G1 are ideal highpass filters, all with cutoff frequency c=/2.

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DSP M l i Fil B k

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DSP: Multirate Filter Banks

X(ej )

2

22

22

22

22

2

2

22

22

22

22

2H0(e

j)X(ej) H1(ej)X(ej)

V0(ej) V1(e

j)

upsampled upsampled

Y0(ej) Y1(e

j)

Y(ej)

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DSP M lti t Filt B k

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DSP: Multirate Filter Banks

Perfect Reconstruction without Ideal Filters

It turns out that we dont need ideal lowpass/highpass filters for this ideato work. As another example, suppose

h0[n] = 1

2([n] + [n 1]) H0(e

j) = cos(/2)ej/2

h1[n] = 12

([n] [n 1]) H1(ej) =j sin(/2)ej/2 =H0(ej())

g0[n] =[n] + [n 1] G0(ej) = 2 cos(/2)ej/2 = 2H0(e

j)

g1[n] = [n] + [n 1] G1(ej) = 2j sin(/2)ej/2 = 2H0(e

j())

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DSP: Multirate Filter Banks

Analysis (part 1 of 2)

Note that the analysis stage filters and downsamples by M= 2. Hence,we can write

V0(ej) =

1

2

X(ej/2)H0(e

j/2) + X(ej(/2))H0(ej(/2))

= ej/4

2

X(ej/2)cos(/4) + X(ej(/2))jsin(/4)

and

V1(e

j

) =

1

2X(e

j/2

)H1(e

j/2

) + X(e

j(/2)

)H1(e

j(/2)

)

=ej/4

2

X(ej/2)j sin(/4) + X(ej(/2))cos(/4)

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DSP: Multirate Filter Banks

Analysis (part 2 of 2)

The synthesis stage upsamples by L= 2 and then filters. Generally, we have

Yi(ej

) =Gi(ej

)Vi(ej2

)

hence we can write

Y0(ej) = 2 cos(/2)ej/2

ej/2

2

X(ej) cos(/2) + X(ej(2))j sin(/2)

= cos(/2)ejX(ej) [cos(/2) +j sin(/2)]

=ejX(ej)

cos2(/2) +j cos(/2) sin(/2)

and

Y1(ej) = 2j sin(/2)ej/2

ej/2

2

X(ej)j sin(/2) + X(ej(2))cos(/2)

= j sin(/2)ej

X(ej

) [j sin(/2) + cos(/2)]=ejX(ej)

sin2(/2) j cos(/2) sin(/2)

It follows then that

Y(ej) =Y0(ej) + Y1(e

j) =ejX(ej)

which means weve recovered our original signal with a one-sample delay.D. Richard Brown III 7 / 8

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DSP: Multirate Filter Banks

Remarks

The filters in the previous examples satisfy the alias cancellationcondition which requires

G0(ej)H0(e

j()) + G1(ej)H1(e

j()) = 0

You can also use the polyphase implementation ideas we developedearlier to reorder the filtering and up/downsampling in the analysisand synthesis operations to save computation:

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