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Linear-Phase FIR Filters

Module by: Ivan Selesnick. E-mail the author

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Summary: (Blank Abstract)

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THE AMPLITUDE RESPONSE

If the real and imaginary parts of H f ω H f ω are given by

H f ω=ℜω+ⅈℑω H f ω ω ω (1)
the magnitude and phase are defined as | H f ω|=ℜω2+ℑω2 H f ω ω 2 ω 2 pω=arctanℑωℜω p ω ω ω so that
H f ω=| H f ω|ⅇⅈpω H f ω H f ω p ω (2)
With this definition, | H f ω| H f ω is never negative and pω p ω is usually discontinuous, but it can be very helpful to write H f ω H f ω as
H f ω=Aωⅇⅈθω H f ω A ω θ ω (3)
where Aω A ω can be positive and negative, and θω θ ω continuous. Aω A ω is called the amplitude response. Figure 1 illustrates the difference between | H f ω| H f ω and Aω A ω .

Figure 1
Figure 1 (ma.png)

A linear-phase phase filter is one for which the continuous phase θω θ ω is linear. H f ω=Aωⅇⅈθω H f ω A ω θ ω with θω=-Mω+B θ ω M ω B We assume in the following that the impulse response hn h n is real-valued.

WHY LINEAR-PHASE?

If a discrete-time cosine signal x 1 n=cos ω 1 n+ φ 1 x 1 n ω 1 n φ 1 is processed through a discrete-time filter with frequency response H f ω=Aωⅇⅈθω H f ω A ω θ ω then the output signal is given by y 1 n=A ω 1 cos ω 1 n+ φ 1 +θ ω 1 y 1 n A ω 1 ω 1 n φ 1 θ ω 1 or y 1 n=A ω 1 cos ω 1 n+θ ω 1 ω 1 + φ 1 y 1 n A ω 1 ω 1 n θ ω 1 ω 1 φ 1 The LTI system has the effect of scaling the cosine signal and delaying it by θ ω 1 ω 1 θ ω 1 ω 1 .

Exercise 1

When does the system delay cosine signals with different frequencies by the same amount?

Solution

  • θωω=constant θ ω ω constant
  • θω=Kω θ ω K ω
  • The phase is linear.

The function θωω θ ω ω is called the phase delay. A linear phase filter therefore has constant phase delay.

WHY LINEAR-PHASE: EXAMPLE

Consider a discrete-time filter described by the difference equation

yn=-0.1821xn+0.7865xn−1−0.6804xn−2+xn−3+0.6804yn−1−0.7865yn−2+0.1821yn−3 y n -0.1821 x n 0.7865 x n 1 0.6804 x n 2 x n 3 0.6804 y n 1 0.7865 y n 2 0.1821 y n 3 (4)
When ω 1 =0.31π ω 1 0.31 , then the delay is −θ ω 1 ω 1 =2.45 θ ω 1 ω 1 2.45 . The delay is illustrated in Figure 2:

Figure 2
Figure 2 (lin1.png)

Notice that the delay is fractional --- the discrete-time samples are not exactly reproduced in the output. The fractional delay can be interpreted in this case as a delay of the underlying continuous-time cosine signal.

WHY LINEAR-PHASE: EXAMPLE (2)

Consider the same system given on the previous slide, but let us change the frequency of the cosine signal.

When ω 2 =0.47π ω 2 0.47 , then the delay is −θ ω 2 ω 2 =0.14 θ ω 2 ω 2 0.14 .

Figure 3
Figure 3 (lin2.png)

note:

For this example, the delay depends on the frequency, because this system does not have linear phase.

WHY LINEAR-PHASE: MORE

From the previous slides, we see that a filter will delay different frequency components of a signal by the same amount if the filter has linear phase (constant phase delay).

In addition, when a narrow band signal (as in AM modulation) goes through a filter, the envelop will be delayed by the group delay or envelop delay of the filter. The amount by which the envelop is delayed is independent of the carrier frequency only if the filter has linear phase.

Also, in applications like image processing, filters with non-linear phase can introduce artifacts that are visually annoying.

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