1. Ideal Filters
One of the reasons why we design a filter is to remove disturbances
Filter
SIGNAL
NOISE
We discriminate between signal and noise in terms of the frequency spectrum

2. Conditions for Non-Distortion
Problem: ideally we do not want the filter to distort the signal we want to recover.
Same shape as s(t), just scaled and delayed.
IDEAL
FILTER
Consequence on the Frequency Response:
constant
linear

3. For real time implementation we also want the filter to be causal, ie.
since
FACT (Bad News!): by the Paley-Wiener Theorem, if h(n) is causal and with finite energy,
ie cannot be zero on an interval, therefore it cannot be ideal.

4. Characteristics of Non Ideal Digital Filters
Positive freq. only
NON IDEAL

5. Two Classes of Digital Filters:
a) Finite Impulse Response (FIR), non recursive, of the form
With N being the order of the filter.
Advantages: always stable, the phase can be made exactly linear, we can approximate any filter we want;
Disadvantages: we need a lot of coefficients (N large) for good performance;
b) Infinite Impulse Response (IIR), recursive, of the form
Advantages: very selective with a few coefficients;
Disadvantages: non necessarily stable, non linear phase.

6. Finite Impulse Response (FIR) Filters
Definition: a filter whose impulse response has finite duration.

7. Problem: Given a desired Frequency Response of the filter, determine the impulse response .
Recall: we relate the Frequency Response and the Impulse Response by the DTFT:
Example: Ideal Low Pass Filter
DTFT

8. Notice two facts:
the filter is not causal, i.e. the impulse response h(n) is non zero for n<0;
the impulse response has infinite duration.
This is not just a coincidence. In general the following can be shown:
If a filter is causal then
the frequency response cannot be zero on an interval;
magnitude and phase are not independent, i.e. they cannot be specified arbitrarily
As a consequence: an ideal filter cannot be causal.

9. Problem: we want to determine a causal Finite Impulse Response (FIR) approximation of the ideal filter.
We do this by
Windowing
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

10. Gibbs phenomenon Cause of 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.
Cause of Gibbs phenomenon:
The FIR filter obtained by truncation can be expressed as:

11. Illustration of the effect of the windowing in frequency domain

12. infinite impulse response
rectangular window
hamming window
infinite impulse response
(ideal)
finite impulse response

13. Effects of windowing and shifting on the frequency response of the filter:
a) Windowing: since then
rectangular window
hamming window

14. For different windows we have different values of the transition region and the attenuation in the stopband:
Rectangular -13dB
Bartlett -27dB
Hanning -32dB
Hamming -43dB
Blackman -58dB
attenuation
with
transition region

15. 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.
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.

16. Fixed Window Functions
Hann:
Hamming:
Blackman:

17. 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

18. 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.

19. 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

20. b) Shifting in time, to make it causal:

21. Frequency Response
Impulse response h(n)

22. Effect of windowing and shifting on the frequency response:
b) shifting: since then
Therefore
See what is
For a Low Pass Filter we can verify the symmetry Then
real for all Then

23. The phase of FIR low pass filter:
Which shows that it is a Linear Phase Filter.
don’t care

24. Example of Design of an FIR filter using Windows:
Specs: Pass Band kHz
Stop Band > 5kHz with attenuation of at least 40dB
Sampling Frequency 20kHz
Step 1: translate specifications into digital frequency
Pass Band
Stop Band
Step 2: from pass band, determine ideal filter impulse response

25. Step 3: from desired attenuation choose the window
Step 3: from desired attenuation choose the window. In this case we can choose the hamming window;
Step 4: from the transition region choose the length N of the impulse response. Choose an odd number N such that:
So choose N=81 which yields the shift L=40.
Finally the impulse response of the filter

26. The Frequency Response of the Filter:

27. A Parametrized Window: the Kaiser Window
The Kaiser window has two parameters:
Window Length
To control attenuation in the Stop Band

28. There are some empirical formulas:
Attenuation in dB
Transition Region in rad
Example:
Sampling Freq. 20 kHz
Pass Band 4 kHz
Stop Band 5kHz, with 40dB Attenuation

29. Then we determine the Kaiser window

30. ideal impulse response
Then the impulse response of the FIR filter becomes
ideal impulse response
with

31. Impulse Response
Frequency Response

32. Example: design a digital filter which approximates a differentiator.
Specifications:
Desired Frequency Response:
Sampling Frequency
Attenuation in the stopband at least 50dB.
Solution.
Step 1. Convert to digital frequency

33. Step 2: determine ideal impulse response
From integration tables or integrating by parts we obtain
Therefore

34. Step 3. From the given attenuation we use the Blackman window
Step 3. From the given attenuation we use the Blackman window. This window has a transition region region of From the given transition region we solve for the complexity N as follows
which yields Choose it odd as, for example, N=121, ie. L=60.
Step 4. Finally the result