Digital filters Desgn

1. Design of Digital Filters
Naila Bibi
BEE (Electronics)
2014 Batch

2. Content
Digital filter introduction
FIR filter design
IRR filter design
Applications
Conclusion

3. What are Digital filters?
In signal processing,
a digital filter is a
system that performs
mathematical
operations on a
sampled, discrete-
time signal to reduce
or enhance certain
aspects of that signal.

4. Why digital filters?
 Digital filters are used for two general purposes
 Separation of signals that have been combined.
 Restoration of signals that have been distorted in some way

5. One of the reasons why we design a filter is to remove disturbances.

)(ns
)(nv
)(nx )()( nsny Filter
SIGNAL
NOISE

6. We discriminate between signal and noise in terms of the frequency spectrum.
F
)(FS
)(FV
0F 0F
0F0F0F

7. Example
Imagine that you have a project for measuring the infant ECG from the
body surface of a mother
•The raw signal recordings will likely be corrupted by the breathing and
heartbeat (ECG) of the mother
•A filter might be used to separate the infant ECG from these combined
signals so that they can be individually analyzed

8. Types of Digital filters
Finite Impulse Response (FIR) filters
Infinite Impulse Response (IIR) filters

9. Comparative analysis of digital Filters
Finite Impulse Response
▪ Implemented non-recursively
▪ No Feedback
▪ Always stable
▪ Simple to implement
▪ Linear phase response in pass-band
▪ More filter coefficients
▪ More memory
▪ More processing power
Infinite Impulse Response
▪ Implement recursively
▪ With Feedback
▪ Stability not guaranteed
▪ Difficult to implement
▪ Non-linear phase response in pass-
band
▪ less filter coefficients than FIR
▪ Less memory
▪ Less processing power

10. The ability to have an exactly linear phase response is the one of the most important of FIR filters
A general FIR filter does not have a linear phase response but this property is satisfied when,
four linear phase filter types,

11. FIR Fillers
 Output is function of the
present input and the past
inputs
 Output does not depend on
the previous outputs
 L+1 is said to be the filter
length

12. IIR Filters
▪ Output is the function of the
present input, the past inputs
and also the past outputs

13. FIR Design Methods
1. Windowing
2. Frequency
Sampling
3. Computer
Aided Design

14. Window Design Method
 Also known as Fourier transform method
 Create an ideal specification of the filter in frequency-domain
 Take Inverse FourierTransform to get the time-domain impulse
response of the filter

15. Steps…
1. Note the specification
i) Pass Band frequency
ii) Pass Band ripple
iii) Stop Band frequency
iv) Stop Band attenuation
2. Find the ideal impulse response; approximate where ever necessary
3. Truncate the ideal impulse response to finite number of samples
4. Choose a suitable window function to ‘smoothen out’ the ideal impulse
response
5. Multiply the ideal-truncated impulse response with the window function
6. The result is the approximated impulse response of the filter, you ideally
wanted to design

16. Example,
-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
|H(w)|
Magnitude Response of Ideal Low Pass Filter
wc-wc
Pass
band
Stop
band
we need to design a
low pass filter.
Pass band cut off at –
Wc to Wc.

17. 

 

otherwise
www
wH cc
0
1
)(
Take the I-DTFT to get the time-domain impulse response of the filter




dwewHnh jwn
d )(
2
1
][


c
c
w
w
jwn
d dwenh )1(
2
1
][


18. c
c
w
w
jwn
d e
jn
nh








1
2
1
][






 


jn
ee
nh
cc jnwjnw
d
2
1
][
j
ee
x
jxjx
2
)sin(








 


j
ee
n
nh
cc jnwjnw
d
2
1
][

 nw
n
nh cd sin
1
][


-∞ < n < ∞ (1)
   c
c
c
cc
d nw
w
nw
nww
nh sinc
sin
][



19. -40 -30 -20 -10 0 10 20 30 40
-0.2
0
0.2
0.4
0.6
0.8
1
time index [n] (-inf to +inf)
hd[n]
There are two ‘major’
problems with it
1. The filter has infinite
number of coefficients
2. It has to be a non-
causal system (system
requires future values)

20. The solution to first problem is to truncate (cut) the impulse response, to a number of
coefficients can be implemented with out too much trouble
N=41
Truncate to
N terms

21. The solution to second problem is to add a delay to the impulse response, so that all coefficients t0
The left of n=0, can be move on the positive time axis.
Move the (N-1)/2Terms to the right
N=41
(N-1)/2 = 20

22. So, the impulse response with the delay of (N-1)/2 samples is given as,
    c
N
Nd wn
n
nh 2
)1(
2
)1(
sin
1
][ 





 
  
 
  c
N
c
N
c
d
wn
wnw
nh
2
1
2
1
sin
][ 





  c
Nc
d wn
w
nh 2
)1(
sinc][ 



23. But…
 With truncation of impulse response, ripples
occur
 In the pass band and stop band
 Due to I.R. suddenly becoming zero
 Going from infinite no. of coefficients to a finite no.
 Known as Gibb’s effect
 Can be mitigated by multiplying I.R. by a
suitable window function
][*][][ nwnhnh d

24. Window Functions

25. The filter coefficients can be calculated as
1)/2-N(,...,2,1n1*][][  nhnh d


6.0
8000
)2400(22

s
c
c
f
f
w
   
 



6.0)2/40(
6.02/40sin6.0
][



n
n
nh
   
)20(
6.020sin1
][



n
n
nh


 
  
 
  c
N
c
N
c
d
wn
wnw
nh
2
1
2
1
sin
][ 





n=0,1,2,…..,20
Where,
h[0] =
h[1] =
h[2] =
h[3] =
h[4] =
h[5] =
h[6] =
h[7] =
h[8] =
h[9] =
h[10] =
h[11] =
h[12] =
h[13] =
h[14] =
h[15] =
h[16] =
h[17] =
h[18] =
h[19] =
h[20] =
0.000000
-0.015933
0.010394
0.011006
-0.018921
0.000000
0.021624
-0.014392
-0.015591
0.027521
0.000000
-0.033637
0.023387
0.026728
-0.050455
0.000000
0.075683
-0.062366
-0.093549
0.302731
0.600000

26. IIR filter Design
▪ From Analog Filters
▪ By approximation of Derivatives
▪ By impulse variance
▪ Bilinear transformation

27. Applications
1. Noise suppression
(a) imaging devices (medical, etc)
(b) bio-signals (heart, brain)
(c) signals stored on analog media
(tapes)

28. Applications continue..
2. Enhancement of selected frequency ranges
(a) equalizers for audio systems (increasing the bass)
(b) Edge enhancement in images

29. Applications continue..
3. Simulation/Modeling
(a) Simulating communication channels
(b) Modeling human auditory system

30. Applications continue..
4. Removal or attenuation of selected frequencies
(a) Removing the DC component of a signal
(b) Removing interferences at a specic frequency,
for example those caused by power supplies

32. Conclusion
Digital Filters
Have a linear phase response in
the band of interest.
Can work for low frequency
signals
Digital filters is not affected by
environmental effects (heat),
There frequency can be easily
adjusted (since it is a program),

33. Thank
you 