1 BIEN425 – Lecture 9 By the end of the lecture, you should be able to: –Describe the properties of ideal filters –Describe the linear / logarithm design specifications for non- ideal filters –Design linear phase FIR filters –Compute the amplitude response of linear phase FIR filters –Decompose any filters with rational transfer function into product of minimum-phase and allpass filters –Represent FIR and IIR filters in direct, parallel and cascade structures

2 Ideal filters Selectively scale the frequency contents of a signal 1 f s /2F p = cut off frequency A(f) f Passband Stopband 0

3 Non-ideal filters Passband Stopband Transition

4 Linear design example (There is an error in the text book) Consider Zeros at z = -1 Poles at z = c To make sure that the filter is stable, we choose |c| < 1 Now assume c = 0.5, and lets evalulate the filter at 2 ends: f = 0: f = f s /2: So what type of filter is it?

5 Let’s find the frequency response H(f) What is the magnitude response A(f):

6 FpFp FsFs 1-  p ss

7 Logarithm design Represent the responses in dB scale Because we want to zoom-in on the dynamics of the transitions and stopbands

8 FpFp FsFs ApAp AsAs

9 Linear phase filters Remember a signal y(k) can be expressed by the following equation where A(f) and  (f) are the amplitude and phase characteristics of a filter respectively. From equation 2.93

10 To preserve the shape integrity of a given input signal, the delay term will have to be independent of frequency In this case, the phase characteristics is as follows Delay: This is called linear phase

11 Nonlinear phase filters The group delay D(f) is defined by: For nonlinear phase filters, D(f) will be a function of frequency (f) and distortion is unavoidable. In this case,  (f) can be  -2  f

12 Linear-phase filter using FIR There is a simple symmetry condition on the coefficients that guarantees a linear phase filter.

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14 Minimum phase filter Every digital filter with rational transfer function can be expressed as a product of two specialized filters: Minimum-phase filter Allpass filter Magnitude response by itself does not provide enough information to completely specify a filter. For an IIR filter with m-zeros, there are 2 m number of distinct filters having the same magnitude response.

15 Define square of magnitude response A 2 (f) Therefore, we can take zero at the reciprocal of b(z)

16 Example

17 Minimum phase filter if and only if all of its zeros lie inside or on the unit cycle.

18 Allpass filter Amplitude response A(f)=1 for all f up to f s /2 Provides phase compensation

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20 Direct form Direct II requires half the storage of Direct I

21 Parallel form

22 Cascade form is less sensitive to coefficient quantization. Cascade form needs less number of fan-outs Cascade form