DIGITAL FILTER STRUCTURE
IMPULSE INVARIANCE METHOD H(z) =
0
) (
n
Z n
n h
H(z) (at z =
e
ST ) = h ( n ) e
STn j T
jw e
re ) r = eT ejw ejT wT
Let S1 = j=> Z1 = eTejT
S2 = 2 )
( T
j
=> Z2 = eTejTj2 eTejT
If the real part is same, imaginary part is differ by integral multiple of 2
T , this is the biggest disadvantage of Impulse Invariance method.
Let HA(S) = finite zero located in the s-plane at s= -a was not converted into a zero in the z-plane at Z =
eaTs, although the zero at s= was placed at z=0.
Desing a Chebyshev LPF using Impulse-Invariance Method.
S1,2 = -473 j 572
S3,4 = -196 j 1380
[The freq response for analog filter we plotted over freq range 0 to 10000. To set the discrete-time freq range (0,
Ts
=
) 88 . 0 707
. 1 1 )(
743 . 0 69
. 1 1
( Z1 Z2 Z1 Z2 k
Methods to convert analog filters into Digital filters:
1. By approximation of derivatives
dt
dx/ t=nTs =
Ts
Ts nTs x nTs
x( ) ( )
S = Ts Z 1 1
Or
Using forward-difference mapping based on first order approximation Z = esTs 1+STs S = Ts
Z1
Using backward- difference mapping is based on first order approximation
STs e
Z1 sTs 1
S ZTs Z 1
= Ts Z 1 1
2 2
dt x
d /t=nTs = t nTs
dt dx dt
d
/
= Ts
Ts nTs x Ts nTs x Ts
Ts nTs x nTs
x( ) ( ) ( ) ( 2 )
= ( ) 2 ( 2 ) ( 2 )
1 using backward difference
Z = 1STs
z is mapped into a circle of radius 0.5, centered at Z=0.5
Using Forward-difference S= Ts
The stable analog filter may be unstable digital filter.
Bilinear Transformation
Provides a non linear one to one mapping of the frequency points on the jw axis in s-plane to those on the unit circle in the z-s-plane.
This procedure also allows us to implement digital HP filters from their analog counter parts.
S = 1
{Using trapezoidal rule y(n)=y(n-1)+0.5Ts[x(n)+x(n-1)]
H(Z)=2(Z-1) / [Ts(Z+1)] }
w such that there is a one to one correspondence between the continuous-time and discrete time frequency points. It is this one to one mapping that allows analog HPF to be implemented in digital filter form.
As in the impulse invariance method, the left half of s-plane maps on to the inside of the unit circle in the z-plane and the right half of s-plane maps onto the outside.
In Inverse relationship is
2 tan w2 Ts
For smaller value of frequency
2 2 2
Cosw Sinw
Ts
=
Ts w w
w w
Ts
4 ....
1 8 ....
2 2
2 3
(B.W of higher freq pass band will tend to reduce disproportionately)
The mapping is linear for small and w. For larger freq values, the non linear compression that occurs in the mapping of to w is more apparent. This compression causes the transfer function at the high freq to be highly distorted when it is translated to the w-domain.
Prewarping Procedure:
When the desired magnitude response is piece wise constant over frequency, this compression can be compensated by introducing a suitable prescaling or prewarping to the
freq scale. scale is converted into * scale.
We now derive the rule by which the poles are mapped from the s-plane to the z-plane.
Let HA(s) = SpTs SpTs
Z
Chebyshev LPF design using the Bilinear Transformation Pass band:
Prewarping values are
p* = The modified specifications are
-1<H(j*)dB 0 for 0 *4484 rad/s Value of N: is determined from
j s*
2Using Impulse Invariance method a value of N=4 was required.
=4.17
Since there are three poles, the angles are
3
Hc(s) =
Pole Mapping At S=S1
Pole Mapping Rules:
Hz(z) = 1-CZ-1 zero at Z=C and pole at Z = 0 Hp(z) = 1
1 1
dZ pole ar Z=d and zero at z=0 C and d can be complex-valued number.
Pole Mapping for Low-Pass to Low Pass Filters
Applying low pass to low pass transformation to Hz(z) we get
HLZ(Z) = 1-c 1
Similarly, HLP(Z)=
K = 0.029 ))
356 . 0 )(
373 . 0 819 . 0 ( 1 ))(
356 . 0 )(
373 . 0 819 . 0 ( 1 )(
356 . 0
* 801 . 0 1
(
j j
UNIT-VI Phase Delay:
()
p
Group Delay:
linear phase filters.
Changes with frequency
g= - =constant.
Type 1 Sequence
Center of Symmetry M=
Type 2 Sequence
h(n) = h(N-1-n)
Center of Symmetry M= 2
1
N half-integer value
H(w) = j MT
N
n
e m n T Cos n
h
1 2
0
) ( )
( 2
The Amplitude spectrum is even symmetric about w=0 & odd symmetric about w= &
both H() is always zero for type 1 & 2 : Constant phase delay and group delay.
Type 3 Sequence
M=
2 1
N integer value
H(w) = j
MT j N
n
e n M T Sin n
h
2 3
0
) (
) ( 2
It shows generalized linear phase of MT 2
and constant group delay of M. The Amplitude spectrum is odd symmetric about w=0 & w= and H(0) & H() are always zero.
(Generalized means () may jump of at 0 if H(ejw) is imaginary.
Type 4 Sequence
H(w) = j j MT
Generalized linear phase and constant group delay of M. The Amplitude spectrum is odd symmetric about w=0 & even symmetric about w= and H(0)=0 always.
For N=even, even Symmetry h(n) = h(N-1-n)
H(ejT) =
==
----Magnitude
N T
1 Linear Phase
Poles & Zeros of linear phase sequences:
The poles of any finite-length sequence must lie at z=0. The zeros of linear phase sequence must occur in conjugate reciprocal pairs. Real zeros at z=1 or z=-1 need not be paired (they form their own reciprocals), but all other real zeros must be paired with their reciprocals. Complex zeros on the unit circle must be paired with their conjugate (that form their reciprocals) and complex zeros anywhere else must occur in conjugate reciprocal quadruples. To identify the type of sequence from its pole-zero plot, all we need to do is check for the presence of zeros at z= and count their number. A type-2 seq must have an odd number of zeros at z=-1, a type-3 seq must have an odd number of zeros at z=-1 and z=1, and type-4 seq must have an odd number of zeros at z=1. The no. of other zeros if present (at z=1 for type=1 and type-2 or z=-1 for type-1 or type-4) must be even.
FIR Filters
Fourier series Method
2 2
F Fs Fs
2 2 2
2
2 Fs
Fs F
2 2
s s
1. Frequency response of a discrete-time filter is a periodi function with period s (sampling freq).
2. From the F.S analysis we know that any periodic function can be expressed as a linear combination of complex exponentials.
Therefore desired freqency response of a discrete time filter can be represented by F.S as
The F.S co-efficient or impulse response samples of filter can be obtained using
h (n) =
clearly if we wish to realize this filter with impulse response h(n), then it must have finite no. of co-efficient, which is equivalent to truncating the infinite expansion of H(ejT), which leads to approximation of H(ejT), which is denoted by
We choose M=
2
However, this filter can’t be physically realizable due to the presence of +ve powers of Z, means that the filter must produce an output that is advanced in time with respect to the i/p.
this difficulty can be overcome by introducing a delay M=
2
be the transfer function of discrete filter that is physically realizable.
Properties:
1. N=2M+1, impulse response co-eff, bi = 0 to 2M.
2. h(n) is symmetric about bM
Ex: M=4
3. The duration of impulse response is Ti = 2MT
4. Its magnitude and time delay function can be found in the following way
) (
)
( e
j Te
j MTH
1e
j TH
) (
)
( e
j TH
1e
j TH
This implies that magnitude response of the filter we have desired approximates the desire magnitude response. The time delay of H(ejw) is a constant M. thus sinusoids of different frequencies are delayed by the same amount as they are processed by the filter, we have designed. Consequently, this is a linear phase filter, which means that it does not introduce phase distortion.
Ex:
Design a LPF (FIR) filter with frequency response
1 ) (ejT
H for c
= 0 for
2 c s
h(n) =
d s e
c c
nT
1
j=
2 s
0cCos ( nT ) d
=
nT
cnT Sin
s
2
= Sin cnT cnT n
Sin Fsn Fs
. 1 2bi = h(i-M)
H(z) =
M i
i i
Z b
2
0
w=
Fs T Fs
T s 1
2 2 2
Ex:
Design LPF that approximate following freq response.
H(F) = 1 0F1000Hz
= 0 else where 1000FFs/2
When the sampling frequency is 8000 SPS. The impulse response duration is to be limited to 2.5ms
Ti = 2MT
M = 10
800
* 1 2
10
* 5 .
2 3 N=21
h(n) =
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OR