Ideal Lowpass Filter (LPF)
where Ωc is the cutoff frequency.
 What is the impulse response of ideal LPF?
Ideal HPF
where Ωc is the cutoff frequency.
 What is the impulse response?
Ideal BPF
where Ωc1 and Ωc2 are the cutoff frequencies.
Ideal BSF
where Ωc1 and Ωc2 are the cutoff frequencies.
FIR Filter
 FIR: finite impulse response (the impulse response
function has a finite length).
 Ex:
 Always stable since
 Long length FIR filter needs lots of delayers.
IIR Filter
 IIR: infinite impulse response (the impulse response
function has infinite length).
 Ex:
 May have stability issue.
 Implemented using feedback loop. (You don not need
infinite number of delayers!)
Distortionless transmission -- Phase
 Phase response of ideal filter was assumed to be zero.
 Ideal filter is physically impossible.
 In practice, linear phase within passband can achieve
distortionless transmission.
 Ex: For a signal with 3 tones:
the following practical filter is applied:
 To look at the output y[k] from the filter, we need t know
the response at each individual frequencies:
where m1, m2, m3 are the slopes of the phase response.
 From previous figure, output is
 Do you see any problem at y[k]?
 If m1≠m2, the output signal y[k] have different phase delays
at different frequencies!
 What does thin mean?  signal is distorted!
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
-2
0
50
100
150
time
200
250
0
50
100
150
time
200
250
Linear phase FIR
 For N-tap FIR with
If the impulse response satisfies:
Symmetrical:
or antisymetrical:
then the frequency response is:
Linear phase FIR types
Example of FIR with linear and nonlinear phase
Example LPF
H ( z )  1  2 z 1  3z 2  2 z 3  z 4
Linear Phase
 H  

H  
9
0.5
0
0.5

 0 .5 
0 .5 
0



 0 .5 

0 .5 
0
H ( z )  1  2 z 1  3z 2  2 z 3  z 4
Non-linear Phase
H  
 H  

7
0.5
0
0.5

 0 .5 
0
0 .5 



 0 .5 
0
0 .5 



Ideal v.s. non-ideal filter
 Ex:
Ideal LPF with linear phase:
Clearly, this IRF has infinite length and is non-causal.
IRF of the ideal LPF:
Truncate the IRF to obtain 2 practical implementations:
141-tap
21-tap
Inverse FT of the two implementation IRF to obtain TF:
Both H1 and H2 are not exactly the ideal LPF.
2. H1 with 141-tap is closer to ILP. H2 with 21-tap shows more
oscillation.
1.
Practical LPF
 Practical filters don’t
have a ‘sharp’ cutoff.
They may also oscillate.
 Ideal filters are non-causal, and cannot be implemented.
 When the Ideal filter impulse response is truncated,
FIR filter results.
 The pass band gain is not constant - includes several
oscillating ripples, referred to as the pass-band ripples.
 The stop band gain is not zero – again it includes
ripples, referred to as the stop band ripples.
If FIR filter length is small:
 More ripples, higher transition bandwidth.
 Low cost, smaller delay
If FIR filter length is large:
 Less ripples, lower transition bandwidth.
 Higher cost, larger delay.
Example
 Impulse response of a 21-tap
FIR filter is given. Estimate
stopband ripple
and
transitionband bandwidth.
 Frequency response:

>> MATLAB Code for plotting the frequency response
>> h = [-0.0014 0.0015 0.0066 0.0081 -0.0059 -0.0330 -0.0411
0.0121 0.1320 0.2619 0.3183 0.2619 0.1320 0.0121
-0.0411 -0.0330 -0.0059 0.0081 0.0066 0.0015 -0.0014] ;
>> stem(k,h,'filled')
>> % Calculating the frequency response
>> [H, w] = freqz(h,1) ;
>> plot(w,20*log10(abs(H))), grid
>> plot(w,angle(H)), grid
h[k ]
 
   3
6
14 15 16
7 8 9 10 11 12 13
k
17 18 19 20
20  log 10  H  
0
20
40
60

 0 .5 
0
0 .5 


Generic filters
 Any linear diff. eq.
can be iteratively solved to obtain:
 Filter realization involves: shifting, adder, and
multiplication.
Hardware to construct digital filter
 Delayer:
 Adder:
 Constant multiplier:
FIR
 Direct form:
 N-th order FIR needs (N-1) delayers.
 Cascade: factorizing TF into multiplication of quadratic
terms.
 N-th order FIR needs (N-1) delayers.
 Linear phase FIR:
Symmetry!
 N is even:
 N is odd:
 (N-1) delayers.
 Transposed form:
Cascade filter:
input  output; reverse
flow direction; source
node  adder;
Reorder the diagram.
IIR
 Direct form I:
when the numerator and
denominator of the TF are,
 Direct form II:
 Interchange the order of N(z) and D(z) from direct form I;
 Delayers are merged.
 Cascade form: factorize N(z) and D(z) into quadratic terms
Q(z) here is from the terms that can’t be factorized as shown previously (for
M≠N ).
 Parallel form:
(expressed as quadratic term to prevent
complex coefficients.)
Choice of structures
 The direct form II (also parallel and cascaded form) is the
so called canonical form and is often used due to simplicity.
 The outputs from all forms should be the same for the
same input. However, due to the finite precision (word
length) of the coefficients, the results may vary.
 Matlab function filterbuilder() provides the coefficients of
all forms. And already considered the finite precision
effect.