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### CHAPTER 7Digital Filter Design

Wang Weilian

wlwang@ynu.edu.cn

School of Information Science and Technology

Yunnan University

Outline

- About Digital Filter Design
- Bilinear Transformation Method of IIR Filter Design
- Design of Lowpass IIR Digital Filters
- Design of Hignpass, Bandpass, and Bandstop IIR Digital Filter
- FIR Filter Design Based on Windowed Fourier Series
- Computer-Aided Design of Digital Filters
- Digital Filter Design Using MATLAB

FIR Filter Design Based on Windowed Series

- Least Integral-Squared Error Design of FIR Filters

In practical application:the desired frequency response is piecewise constant with sharp transitions between bands.

Aim:Find a finite –duration impulse response sequence of the length 2M+1 whose DTFT

approximates the desired DTFT In some sense.

one commonly used approximation criterion is to minimize the integral-squared error.

FIR Filter Design Based on Windowed Series

Integral-squared error

The integral-squared error is minimum when = for .

FIR Filter Design Based on Windowed Series

- Impulse Response of Ideal Filters

Four commonly used frequency selective filters are the lowpass,highpass,bandpass,bandstop filters.

- Example:lowpass filter

zero-phase frequency response

The corresponding impulse response

so the impulse response is doubly infinite,not absolutely summable,and therefore unrealizable.

FIR Filter Design Based on Windowed Series

By setting all impulse response coefficient outside the range equal to zero,we arrival at a finite-length noncausal approximation of length

,which when shifted to the right yield the coeffcients of a causal FIR lowpass filter:

FIR Filter Design Based on Windowed Series

- Gibbs phenomenon

The causal FIR filter obtained by simply truncating the impulse response coefficients of the ideal filters exhibit an oscillatory behavior in their respective magnitude responses.which is more commonly referred to as the Gibbs phenomenon.

FIR Filter Design Based on Windowed Series

- Cause of Gibbs phenomenon:

The FIR filter obtained by truncation can be expressed as:

FIR Filter Design Based on Windowed Series

- Illustration of the effect of the windowing in frequency domain

FIR Filter Design Based on Windowed Series

The window used to achieve simple truncation of the ideal filter is rectangular window:

So two basic reason of the oscillatory behavior:

(1)the impulse response of a ideal filter is infinitely long and not absolutely summable.

(2)the rectangular window has an abrupt transition to zero.

FIR Filter Design Based on Windowed Series

- How to reduce the Gibbs phenomenon?

(1 )using a window that tapers smoothly to zero at each end.

(2)providing a smooth transition from the passband to the stopband.

FIR Filter Design Based on Windowed Series

- Two important parameters:

(1)main lobe width.

(2)relative sidelobe level.

The effect of window function on FIR filter design

(1) the window have a small main lobe width will ensure a fast transition from the passband to the stopband.

(2)the area under the sidelobes small will reduce the ripple

FIR Filter Design Based on Windowed Series

- Designing an FIR filter

(1)select a window above mentioned.

(2)get

(3)determine the cutoff frequency by setting:

(4)M is estimated using ,the value of the constant c is obtain from table given.

FIR Filter Design Based on Windowed Series

- Adjustable Window Functions

Windows have been developed that provide control over ripple by means of an additional parameter.

(1)Dolph-Chebyshev window

(2)Kaiser window

FIR Filter Design Based on Windowed Series

- Impulse Response of FIR Filters with a Smooth Transition

--One way to reduce the Gibbs phenomenon.

The simplest modification to the zero-phase lowpass filter specification is to provide a transition band between the passband and stopband responses and to connect these two with a first order spline function .

Computer-Aided Design of Digital Filter

- Two specific design approaches based in iterative potimization techniques.

The aim is to determine iteratively the coefficients of the digital transfer function so that the difference between and for all value of over closed subintervals of is minimized ,and usually the difference is specified as a weighted error function given by:

Computer-Aided Design of Digital Filter

- Chebyshev criterion

--to minimize the peak absolute value of the weighted error

- Least-p criterion

--to minimize the integral of pth power of the weighted error function

Computer-Aided Design of Digital Filter

- Design of Equiripple Linear-Phase FIR Filter

The frequency response of a linear-phase FIR filter is:

The weighted error function in this case involves the amplitude response and is given by

Computer-Aided Design of Digital Filter

- Type 1 linear-phase FIR filter

The amplitude response is :

It can be rewrite using the notation in the form

Where

Computer-Aided Design of Digital Filter

- Type 2 linear-phase FIR filter

The amplitude response is :

It can be rewrite in the form:

Where

Computer-Aided Design of Digital Filter

- Type 3 linear-phase FIR filter

The amplitude response is :

It can be rewrite in the form:

Computer-Aided Design of Digital Filter

- Type 4 linear-phase FIR filter

The amplitude response is :

It can be rewrite in the form:

Computer-Aided Design of Digital Filter

- The amplitude response for all four types of linear-phase FIR filters can be expressed in the form
- Then the we modify the form of the weight approximation function as:

Computer-Aided Design of Digital Filter

Using the notions and

we can rewrite it as:

Then we determine the coefficients to minimize the peak absolute value of the weighted approximation error over the specified frequency bands

Computer-Aided Design of Digital Filter

- Alternation Theorem

The amplitude function is the best unique aproximation of the desired amplitude response obtained by minimizing the peak absolute valu

of if and only if there exist at least

extremal angular frequencies, ,in a closed subset R of the frequency range

such that and

with for all in the range

Digital Filter Design Using Matlab

- IIR Digital Filter Design Using Matlab
- Steps:(1)determine the filter order N and the frequency scaling factor Wn .

[N,Wn]=buttord(Wp,Ws,Rp,Rs)

[N,Wn]=cheb1ord(Wp,Ws,Rp,Rs)

[N,Wn]=cheb2ord(Wp,Ws,Rp,Rs)

[N,Wn]=ellipord(Wp,Ws,Rp,Rs)

Where Wp=2Fp/FT and Ws= 2Fs/FT .

Digital Filter Design Using Matlab

- (2)determine the coefficients of the transfer function.

[b,a]=butter(N,Wn)

[b,a]=cheby1(N,Rp,Wn)

[b,a]=cheby2(N,Rs,Wn)

[b,a]=ellip(N,Rp,Rs,Wn)

Digital Filter Design Using Matlab

- FIR Digital Filter Design Using Matlab
- Steps(1).estimate the filter order from the given specification.

remezord,kaiserord

- (2)determine the coefficient of the transfer function using the estimated order and the filter specification.

remez

Digital Filter Design Using Matlab

- FIR Digital Filter Order Estimation Using Matlab

[N,fpts,mag,wt]=remezord(fedge,mval,dev)

[N,fpts,mag,wt]=remezord(fedge,mval,dev,FT)

For FIR filter design using the Kaiser window,the window order should be estimated using kaiserord

[N,Wn,beta,ftype]=kaiserord(fedge,mval,dev)

[N,Wn,beta,ftype]=kaiserord(fedge,mval,dev,FT)

C=kaiserord(fpts,mval,dev,FT,’cell’)

Digital Filter Design Using Matlab

- Equiripple Linear-phase FIR Design Using Matlab

--emplying the Parks-McClellan algorithm.

b=remez(N,fpts,mag)

b=remez(N,fpts,mag,wt)

b=remez(N,fpts,mag,’ftype’)

b=remez(N,fpts,mag,wt,’ftype’)

Digital Filter Design Using Matlab

- FIR equiripple lowpass filter of Example 7.27 for N=28

Digital Filter Design Using Matlab

- Gain response of the FIR equiripple bandpass filter of Example 7.28.

Digital Filter Design Using Matlab

- Window-based FIR Filter Design Using Matlab
- Steps:

(1)estimate the order of the FIR filter.

(2)select the type of the window and compute its coefficient.

(3)compute the desired impluse response of the ideal filter.

Digital Filter Design Using Matlab

- Window Generation

W=blackman(L)

W=hamming(L)

W=hanning(L)

W=chebwin(L,Rs)

W=kaiser(L,beta)

Digital Filter Design Using Matlab

- Filter Design

fir1 is used to design conventional lowpass,highpass, bandpass,bandstop and multiband FIR filter.

b=fir1(N,Wn)

b=fir1(N,Wn,’ftype’)

b=fir1(N,Wn,window)

b=fir1(N,Wn,’ftype’window)

b=fir1(……,’noscale’)

Digital Filter Design Using Matlab

- A example of a conventional lowpass FIR filter

Digital Filter Design Using Matlab

- Filter Design

fir2 is employed to design FIR filters with arbitarily shaped magnitude response.

b=fir2(N,f,m)

b=fir2(N,f,m,window)

b=fir2(N,f,m,npt)

b=fir2(N,f,m,npt,window)

b=fir2(N,f,m,npt,lap,window)

Digital Filter Design Using Matlab

- A Examples of multilevel filter

--Magnitude response of the multilevel filter designed with fir2

Digital Filter Design Using Matlab

- Least-squares Error FIR Filter Design Using Matlab

firls –to design any type of multiband linear-phase FIR filter based on the least-squares method

b=firls(N,fpts,mag)

b=firls(N,fpts,mag,wt)

b=firls(N,fpts,mag,’ftype’)

b=firls(N,fpts,mag,wt,’ftype’)

Digital Filter Design Using Matlab

- A example of the linear-phase FIR lowpass filter

--Gain response of the linear-phase FIR lowpass filter

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