digital filter design_2nd

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DIGITAL FILTER DESIGN

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Spectral Transformations ofIIR Digital Filters

• Objective - Transform a given lowpass digital transfer function GL(z) to another digital transfer function GD(ẑ) that could be a lowpass, highpass, bandpass or bandstop filter.

• z-1 has been used to denote the unit delay in the prototype lowpass filter GL(z) and ẑ-1 to denote the unit delay in the transformed filter GD(ẑ) to avoid confusion.

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• Unit circles in z- and ẑ-planes defined by

• Transformation from z-domain to ẑ-domain given by

• Then

ˆjˆ, z = ejz e

( )ˆz F z

ˆ ˆD LG z G F z

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• From , thus , hence

• Therefore 1/F(ẑ) must be a stable allpass function

( )ˆz F z ˆz F z

1, if 1

ˆ 1, if 1

1, if 1

z

F z z

z

*

1

ˆ11, 1

ˆ ˆ

Ll

ll l

z

F z z

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Lowpass-to-LowpassSpectral Transformation

• To transform a lowpass filter GL(z) with a cutoff frequency ωc to another lowpass filter GD(ẑ) with a cutoff frequency , the transformation is

• On the unit circle we have

Which yields

ˆc

1 ˆ1 1

ˆ ˆz

zF z z

ˆ

ˆ1

jj

j

ee

e

1ˆtan 2 tan 2

1

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• Solving we get

• Example - Consider the lowpass digital filter

which has a passband from dc to 0.25π with a 0.5 dB ripple.

• Redesign the above filter to move the passband edge to 0.35π.

ˆsin 2

ˆsin 2c c

c c

31

1 1 2

0.0662 1

1 0.2593 1 0.6763 0.3917L

zG z

z z z

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• Here

• Hence, the desired lowpass transfer function is

sin 0.050.1934

sin 0.3

11

1

ˆ 0.1934

ˆ1 0.1934

ˆ zD L zz

G z G z

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• The lowpass-to-lowpass transformation

can also be used as highpass-to-highpass, bandpass-to-bandpass and bandstop-to-bandstop transformations.

1 ˆ1 1

ˆ ˆz

zF z z

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Lowpass-to-HighpassSpectral Transformation

• Desired transformation

• The transformation parameter α is given by

where ωc is the cutoff frequency of the lowpass filter and is the cutoff frequency of the desired highpass filter.

11

1

ˆ

ˆ1

zz

z

ˆcos 2

ˆcos 2c c

c c

ˆc

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• Example - Transform the lowpass filter

with a passband edge at 0.25π to a highpass filter with a passband edge at 0.55π.

• Here, • The desired transformation is

31

1 1 2

0.0662 1

1 0.2593 1 0.6763 0.3917L

zG z

z z z

cos 0.4 cos 0.15 0.3468

11

1

ˆ 0.3468ˆ1 0.3468

zz

z

11


• The desired highpass filter is

11

1

ˆ 0.3468

ˆ1 0.3468

ˆ zD L zz

G z G z

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• The lowpass-to-highpass transformation can also be used to transform a highpass filter with a cutoff at ωc to a lowpass filter with a cutoff at .

• And transform a bandpass filter with a center frequency at ω0 to a bandstop filter with a center frequency at

ˆc

0ˆ .

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Lowpass-to-BandpassSpectral Transformation


2 1

1

2 1

2 1ˆ ˆ

1 11 2

ˆ ˆ 11 1

z zz

z z

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• The parameters α and β are given by

where ωc is the cutoff frequency of the lowpass filter, and and are the desired upper and lower cutoff frequencies of the bandpass filter.

2 1

2 1

ˆ ˆcos 2

ˆ ˆcos 2c c

c c

2 1ˆ ˆcot 2 tan 2c c c

1ˆc 2ˆc

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• Special Case - The transformation can be simplified if

• Then the transformation reduces to

where with denoting the desired center frequency of the bandpass filter.

2 1ˆ ˆc c c

11 1

1

ˆˆ

ˆ1

zz z

z

0ˆcos 0̂

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Lowpass-to-BandstopSpectral Transformation


2 1

1

2 1

2 1ˆ ˆ

1 11 2

ˆ ˆ 11 1

z zz

z z

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Lowpass-to-BandstopSpectral Transformation

• The parameters α and β are given by

where ωc is the cutoff frequency of the lowpass filter, and and are the desired upper and lower cutoff frequencies of the bandstop filter.

2 1

2 1

ˆ ˆcos 2

ˆ ˆcos 2c c

c c

2 1ˆ ˆtan 2 tan 2c c c

1ˆc 2ˆc

digital filter design_2nd

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