Practical Frequency-Selective Digital Filter Design

Practical Considerations in Digital Filter Design

Digital Filter Design I

Practical Frequency-Selective Digital Filter Design

Desired filter characteristics are specified in the frequency domain in terms of desired magnitude and phase response of the filter; i.e., H(ω) is specified.

Passband

Dr. Deepa Kundur

Transition band

Passband edge frequency Stopband edge frequency

University of Toronto Stopband

I

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Practical Considerations in Digital Filter Design

Filter design involves determining the coefficients of a causal FIR or IIR filter that closely approximates the desired frequency response specifications.

Dr. Deepa Kundur (University of Toronto) Practical Frequency-Selective Digital Filter Design

Practical Frequency-Selective Digital Filter Design

FIR versus IIR Filters

Linear Phase

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Practical Considerations in Digital Filter Design

Passband ripple

Passband ripple Stopband ripple Passband edge frequency Stopband edge frequency

I

FIR filters: normally used when there is a requirement of linear phase I

FIR filter with the following symmetry is linear phase: h(n) = ±h(M − 1 − n)

I

Q: What is linear phase?

n = 0, 1, 2, . . . , M − 1

IIR filters: normally used when linear phase is not required and cost effectiveness is needed I

I

A: The phase is a straight line in the passband of the system.

IIR filter has lower sidelobes in the stopband than an FIR having the same number of parameters if some phase distortion is tolerable, an IIR filter has an implementation with fewer parameters requiring less memory and lower complexity

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Practical Frequency-Selective Digital Filter Design

Practical Considerations in Digital Filter Design

Practical Frequency-Selective Digital Filter Design

Practical Considerations in Digital Filter Design

Linear Phase

Linear Phase

Example: linear phase (all pass system) I Group delay is given by the negative of the slope of the line (more on this soon).

Example: linear phase (all pass system) I Phase wrapping may occur, but the phase is still considered to be linear.

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Practical Considerations in Digital Filter Design

Dr. Deepa Kundur (University of Toronto) Practical Frequency-Selective Digital Filter Design

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Practical Considerations in Digital Filter Design

Linear Phase

Linear Phase

Example: linear phase (high pass system) I Discontinuities at the origin still correspond to a linear phase system.

Example: linear phase (low pass system) I Linear characteristics only need to pertain to the passband frequencies only.

Passband

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Practical Frequency-Selective Digital Filter Design

Practical Considerations in Digital Filter Design

Practical Frequency-Selective Digital Filter Design

DTFT Theorems and Properties

Practical Considerations in Digital Filter Design

Group Delay Therefore,

Recall, Property Notation:

Linearity: Time shifting: Time reversal Convolution: Correlation:

Time Domain x(n) x1 (n) x2 (n) a1 x1 (n) + a2 x2 (n) x(n − k) x(−n) x1 (n) ∗ x2 (n) rx1 x2 (l) = x1 (l) ∗ x2 (−l)

Wiener-Khintchine:

rxx (l) = x(l) ∗ x(−l)

Frequency Domain X (ω) X1 (ω) X1 (ω) a1 X1 (ω) + a2 X2 (ω) e −jωk X (ω) X (−ω) X1 (ω)X2 (ω) Sx1 x2 (ω) = X1 (ω)X2 (−ω) = X1 (ω)X2∗ (ω) [if x2 (n) real] Sxx (ω) = |X (ω)|2

group delay

Y (ω) X (ω) ∠H(ω) = Φ(ω) H(ω) =

I

e −jωn0

=

−ωn0 = −ω · group delay

group delay ≡ −

Dr. Deepa Kundur (University of Toronto) Practical Frequency-Selective Digital Filter Design

Practical Frequency-Selective Digital Filter Design

=

In general (even for nonlinear phase systems),

among others . . .

I

F

n0 ) ←→ Y (ω) = X (ω)e −jωn0 |{z}

y (n) = x(n −

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Practical Considerations in Digital Filter Design

dΦ(ω) dω

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Practical Considerations in Digital Filter Design

Signal Magnitude versus Signal Phase

Linear phase filters maintain the relative positioning of the sinusoids in the filter passband. This maintains the structure of the signal while removing unwanted frequency components.

Q: Why is linear phase important? Q: What can happen when there is loss of phase information? -3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

n

Passband input

linear in passband

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output

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Practical Frequency-Selective Digital Filter Design

Practical Considerations in Digital Filter Design

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Practical Frequency-Selective Digital Filter Design

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Practical Considerations in Digital Filter Design

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Practical Frequency-Selective Digital Filter Design

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Practical Considerations in Digital Filter Design

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Practical Considerations in Digital Filter Design

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Practical Frequency-Selective Digital Filter Design

Practical Considerations in Digital Filter Design

Practical Frequency-Selective Digital Filter Design

Practical Considerations in Digital Filter Design

Signal Magnitude versus Signal Phase

A: To maintain the original “structure” of a signal in the passband frequency range, linear phase (or close to linear phase) is required.

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Practical Considerations in Digital Filter Design

Practical Frequency-Selective Digital Filter Design

Linear Phase FIR Filters I

I

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Practical Considerations in Digital Filter Design

Linear Phase FIR Filters: Example

As mentioned previously, FIR filters with the following symmetry are linear phase: h(n) = ±h(M − 1 − n)

Dr. Deepa Kundur (University of Toronto) Practical Frequency-Selective Digital Filter Design

Q: Show that h(n) = δ(n) − δ(n − 1) is linear phase by determining the associated phase and group delay. Note: M = 2 and h(n) = −h(1 − n) = −h(M − 1 − n) for n = 0, 1.

n = 0, 1, 2, . . . , M − 1

h(n)

Note that this means that

1

h(n) = +h(M − 1 − n)

1 -7 -6 -5 -4 -3 -2 -1 0

for n = 0, 1, 2, . . . , M − 1, or

2 3 4 5 6 7

n

-1

h(n) = −h(M − 1 − n) For n = 0, h(0) = −h(1 − 0) = 1 and n = 1, h(1) = −h(1 − 1) = −1.

for n = 0, 1, 2, . . . , M − 1. Dr. Deepa Kundur (University of Toronto) Practical Frequency-Selective Digital Filter Design

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Practical Frequency-Selective Digital Filter Design

Practical Considerations in Digital Filter Design

Practical Frequency-Selective Digital Filter Design

Linear Phase FIR Filters: Example

Practical Considerations in Digital Filter Design

Linear Phase FIR Filters: Example

Q: Show that h(n) = δ(n) − δ(n − 1) is linear phase by determining the associated phase and group delay. Note: This system corresponds to: y (n) = = = first difference ⇔

H(ω) =

x(n) ∗ h(n) = x(n) ∗ [δ(n) − δ(n − 1)] x(n) ∗ δ(n) − x(n) ∗ δ(n − 1) x(n) − x(n − 1) (first difference system) dst-time derivative ⇒ highpass filter

Dr. Deepa Kundur (University of Toronto) Practical Frequency-Selective Digital Filter Design

Practical Frequency-Selective Digital Filter Design

∞ X

h(n)e −jωn

n=−∞ −jω·0

= 1·e + (−1) · e −jω·1
= 1 − e −jω = e −jω/2 e jω/2 − e −jω/2 = e −jω/2 · 2j sin(ω/2) = 2je −jω/2 sin(ω/2)

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Practical Considerations in Digital Filter Design

Dr. Deepa Kundur (University of Toronto) Practical Frequency-Selective Digital Filter Design

Practical Frequency-Selective Digital Filter Design

Linear Phase FIR Filters: Example

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Practical Considerations in Digital Filter Design

Linear Phase FIR Filters: Example

Note: |H(ω)| = |2je −jω/2 sin(ω/2)| = |2| · |j| · |e −jω/2 | · | sin(ω/2)| = 2 · 1 · 1 · | sin(ω/2)| = 2| sin(ω/2)|

Φ(ω) = ∠2je −jω/2 sin(ω/2) −jω/2 = |{z} ∠2 + ∠j + ∠e + |{z} | {z } =0



2

=  =

linear in

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=π/2

=−ω/2

 =

∠ sin(ω/2) | {z } 0 π

0