Linear Phase

7/31/2019 Linear Phase

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Linear Phase Filters

VOCAL Technologies LTD

October 25th, 2012

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I. Introduction

Linear phase filters are practical realizations of zero phase filters. A zero

phase filter is a filter that has symmetry about the origin. Thus when using a

zero phase filter, one must take negative indices into account, which is rather

tedious and is prone to error. To make things easier, we instead make the filter

symmetric about some non-zero point. What results is a linear phase filter

shown below:

Figure 1: A Linear Phase Filter The Hamming Window

Linear phase filters are important for audio filter design because passing a

signal through a linear phase filter will delay all of the frequency components

by the same amount, thus leaving the structure of the signal intact, which will

preserve speech intelligibility. To understand how phase effect delay, its quite

common to start by looking at the frequency response of the ideal delay system

h[n] = [n nd]:

H(ej) =

+

k=

[n nd]ejk = ejnd (1)

This behavior is due to the sampling property of the delta function. We

can see that the phase is then:

H(ej) = arg[ejnd ] = nd (2)

Which is a linear function of frequency. Thus, time delay and linear phase

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are inexorably linked. If you still dont believe, consider what happens when we

put a signal x through the ideal delay system. The output is:

y[n] = (x|h)[n] = x[n nd] (3)

This has frequency response:

Y(ej) = X(ej)H(ej) = X(ej)ejnd (4)

As the equation shows, regardless of what the frequency response of the

input x is, it is shifted linearly by nd, while the time domain signal is shifted

by nd. Again, time delay and linear phase are inexorably linked. With this in

mind, it will be useful to examine some different types of filters that will giveus a linear phase response. First, we will examine the four types of FIR filters

[1].

II. Linear Phase FIR Filters

The four types of linear phase FIR filters are appropriately named Type

I-IV. These filters are categorized by whether they have an even or odd length,

and whether they are symmetric or antisymmetric. In all cases, we will define

the center of symmetry to be M2

where M 1 is the length of the filter. Since

the filters will have essentially the same shape on either side of the center ofsymmetry, we can think of each as a filter delayed by M

2. The examples follow

from [1].

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Type I FIR Filters

Figure 2: Type I FIR Filter

As the above figure shows, Type I FIR Filters are odd length filters with a

symmetric impulse response given by:

h[n] = h[M n] 0 n M (5)

Most generally, these filters can be represented as:

H(ej) = ejM

2

M

2

k=0

a[k]cos(k) (6)

Where:

a[0] = h[M

2]

a[k] = 2h[M

2 k] 1 k

M

2(7)

For the example above, the frequency response is:

H(ej) =

6

k=0

h[n]ejwn

= 0 ejw0 + 1 ejw1 + 2 ejw2 + 3 ejw3 + 2 ejw4 + 1 ejw5 + 0 ejw6

= ej3w(ej2w + 2 ejw + 3 + 2 ejw + ej2w)

= ej3w(2(ejw + ejw) + (ej2w + ej2w) + 3)

= ej3w(4cos(w) + 2cos(2w) + 3) (8)

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Type II FIR Filters

Figure 3: Type II FIR Filter

As the above figure shows, Type II FIR Filters are even length filters with

a symmetric impulse response. Most generally, these filters can be represented

as:

H(ej) = ejM

2

M+1

2

k=1

b[k]cos[(k 1

2)] (9)

Where:

a[k] = 2h[M+ 1

2 k] 1 k

M+ 1

2(10)

For the example above, the frequency response is:

H(ej) =

5

k=0

h[n]ejwn

= 1 ejw0 + 2 ejw1 + 3 ejw2 + 3 ejw3 + 2 ejw4 + 1 ejw5

= ej52w(ej

52w + 2 ej

32w + 3 ej

12w + 3 ej

12w + 2 ej

32w + ej

52w)

= ej52w((ej

52w + ej

52w) + 2(ej

32w + ej

32w) + 3(ej

12w + ej

12w)

= ej52w(2cos(

5

2w) + 4cos(

3

2w) + 6cos(

1

2w)) (11)

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Type III FIR Filters

Figure 4: Type III FIR Filter

As the above figure shows, Type III FIR Filters are odd length filters with

an antisymmetric impulse response given by:

h[n] = h[M n] 0 n M (12)

Most generally, these filters can be represented as:

H(ej) = jejM

2

M

2

k=1

c[k]sin(k) (13)

Where:

c[k] = 2h[M

2 k] 1 k

M

2(14)

Similarly to the Type I filter, the frequency response of the above example

is:

H(ej) =6

k=0

h[n]ejwn

= ej3w+

2 (4sin(w) + 2sin(2w) + 3) (15)

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Type IV FIR Filters

Figure 5: Type IV FIR Filter

As the above figure shows, Type IV FIR Filters are even length filters with an

antisymmetric impulse response. Most generally, these filters can be represented

as:

H(ej) = jejM

2

M+1

2

k=1

d[k]sin[(k 1

2)] (16)

Where:

d[k] = 2h[M+ 1

2 k] 1 k

M+ 1

2(17)

Similarly to the Type II filter, the frequency response of the above example

is:

H(ej) =

6

k=0

h[n]ejwn

= ej52w+

2 (2sin(5

2w) + 4sin(

3

2w) + 6sin(

1

2w)) (18)

With analogy to (4), it is clear that all of these filters have linear phase.

From this discussion, it should be clear to you which FIR filters will be linear

phase, and which will not be. Similarly, by shifting in the time domain, you cancreate a linear phase filter from a zero phase one.

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III. Linear Phase IIR Filters

IIR Filters may be made linear phase by using what is called bi-directional

filtering. In bi-directional filtering, the signal of interest is first filtered via

convolution with the IIR filter from left-to-right, and then the output is filtered

via the IIR filter going from right-to-left for instance. In this manner, filters

that are generally not linear phase like IIR filters can be effectively transformed

to filters that are.

Of course, in practice, this method demands a great deal of delay because

not only do you have to wait for the signal values to create the IIR filter, but

you also need to filter the signal twice. However, if your FIR filter is sufficiently

long, using a much lower order IIR filter may be worth the extra computational

overhead.

References

[1] A. V. Oppenheim, R. W. Schafer, Transform Analysis of Linear Time-

Invariant Systems, in Discrete Time Signal Processing, 2nd ed. Upper Saddle

River, NJ: Prentice-Hall Inc., 1999, ch. 5, sec. 7, pp. 298300.

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