1. Fourier Series FIR 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

2. 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.

3. FIR Filter Design Based on Windowed Series
Integral-squared error
The integral-squared error is minimum when = for

4. 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.

5. 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:

6. 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.

7. FIR Filter Design Based on Windowed Series
Cause of Gibbs phenomenon:
The FIR filter obtained by truncation can be expressed as:

8. FIR Filter Design Based on Windowed Series
Illustration of the effect of the windowing in frequency domain

9. 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.

10. 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.

11. FIR Filter Design Based on Windowed Series
Fixed Window Functions
Hann:
Hamming:
Blackman:

12. 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

13. 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.

14. 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

15. 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 .

16. 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:

17. 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

18. 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

19. 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

20. Computer-Aided Design of Digital Filter
Type 2 linear-phase FIR filter
The amplitude response is :
It can be rewrite in the form:
Where

21. Computer-Aided Design of Digital Filter
Type 3 linear-phase FIR filter
The amplitude response is :
It can be rewrite in the form:

22. Computer-Aided Design of Digital Filter
Type 4 linear-phase FIR filter
The amplitude response is :
It can be rewrite in the form:

23. 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:

24. 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

25. 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

26. 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 .

27. 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)

28. 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

29. 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’)

30. 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’)

31. Digital Filter Design Using Matlab
FIR equiripple lowpass filter of Example 7.27 for N=28

32. Digital Filter Design Using Matlab
Gain response of the FIR equiripple bandpass filter of Example 7.28.

33. 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.

34. Digital Filter Design Using Matlab
Window Generation
W=blackman(L)
W=hamming(L)
W=hanning(L)
W=chebwin(L,Rs)
W=kaiser(L,beta)

35. Digital Filter Design Using Matlab
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’)

36. Digital Filter Design Using Matlab
A example of a conventional lowpass FIR filter

37. Digital Filter Design Using Matlab
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)

38. Digital Filter Design Using Matlab
A Examples of multilevel filter
--Magnitude response of the multilevel filter designed with fir2

39. 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’)

40. Digital Filter Design Using Matlab
A example of the linear-phase FIR lowpass filter
--Gain response of the linear-phase FIR lowpass filter