Designing filters involves the
mapping and transformation of well-known continuous time filter designs to discrete time ones
Impulse Invariance means that the
Impulse response of the digital filter will be similar to that of the continuous time filter
How can the impuulse response of a CT Filter be similar to that of a DT Filter or Digital filters
by sampling of the impulse response of the CT Filter
Impulse invariance maps the _ to -
S domain to the Z domain
Design Procedure of Digital filters when using Impulse invariance involves
Starting with an analogue prototype or Transfer Function H(s)
Find the inverse Laplace Transform to obtain analogue prototype impulse response h(t)
Sample the impulse response h[n] = h[nT]
The desired transfer function is found by taking the z transform of h[n]
Result of Sampling the analogue prototype impulse response gives
an impulse response of the discrete filter h[n] that is identical to the analogue filter at t = nT
sampling in the time domain will cause frequency aliasing
Impulse response transformation involves
the conversion of t→ n = nT
Frequency Response Transformation involves
the conversion of ώ → Ω = ώT
Design Limitations of the Impulse Invariance is
The frequency response of the CT filter has to be bandlimited for aliasing not to occur
High pass and band -stop filter design is not possible with impulse invariance
Practical continuous time filters are not exactly band limited and some aliasing occurs
In practise to compensate for aliasing the continuous time filter may be overdesigned ( lower cut- off, higher attenuation in stop-band)
Impulse invariance indirectly maps the
s-plane to the z-plane through sampling of the impulse response
Bilinear Transformation
applies a non-linear mapping of the analogue frequency axis to the digital frequency one
IIR filter design steps involve
specification of the continuous -time prototype low-pass filter
frequency transformation of the analogue prototype low-pass filter
transformation of the CT filter to DT filter
Types of Analogue prototype filters
Butterworth Analogue filter
Chebychev Analogue Filter
Elliptic analogue filter
Butterworth Analogue Filter
Maximal flatness at Ω = 0 and Ω = ∞
Maximum distortion at Ω = Ω_c
Not a very sharp cut-off
Chebychev filter specification
Equiripple passband (type I)
Equiripple stopband (type II)
sharper cut off compared to butterworth
lower order filter needed to satisfy the requirements
if the chebychev and butterwort filter have the same design specification for passband and stop band edge , pass band ripple and minimum attenuation, Who has a better performance
Chebychev because it has a sharper transition band
non linear mapping involves
−∞≤ώ≤∞ → −π≤ώ≤π
properties of bilinear transformation
stable CT filter maps to a stable DT filter i.e half of s plane maps to inside the unit circle
the jώ axis maps on the unit circle
s and z relationship is given as
s = 2/T[(1-z¹)/(1-z¹)]
where z¹ = z^-1
ώ relationship to Ω
ώ = 2/T tan(Ω/2)
Frequency Warping
non-linear compression of the frequency axis
to solve the problem of frequency warping or to compensate the warping effect
pre-warp the frequency design specifications of the DTfilter using
ώ=2tan(Ω/2)/T
Steps in using pre-warping to get desired analogue frequency
The desired stop- band frequency is given
Pre-warp to get analogue filters requires stop band frequency
bilinear transformation is applied s → z
results in a digital filter with the desired stop-band
Design Procedure using Bilinear Transformation
Incorporate frequency axis pre-warping as appropriate
select analogue filter prototype H(s) to satisfy design spec (Butterworth, Chebychev)
Apply Frequency Transformation ( changing from lowpass to any type)
Evaluate H(s) and apply bilinear transform to obtain H(z)
Bilinear Transformation Disadvantages
Frequency axis warping
Distortion of phase characteristics
Distortion of time domain characteristics
Direct Optimisation
Specify the design objectives
Choose filter structure and paramters
Select optimisation tool to find the optimal or sub optimal set of filter parameters