LTI filters are characterized by their impulse response. The two broad categories of LTI systems are those with finite impulse responses (FIR) and those with infinite impulse responses (IIR). LTI systems are frequently characterized by linear constant-coefficient difference equations which look as follows:
∑k=0Naky[n−k]=∑k=0Mx[n−k].
Definition 13
The system function H(z) is the z-transform of the impulse response of the system. For LCCDE's, it is a ratio of polynomials in z−1.
We can systematically analyze these by drawing a vector from e−jω to each dk or ck and analyze each one individually. For example, if we look at one pole in the magnitude response
∣1−dke−jω∣=∣ejω−dk∣=∣vk∣.
In general, the effects of poles and zeros on each of these quantities is described by the following table.
Magnitude Response
Phase Response
Group Delay
Poles
Increase
Phase Lag
Increase
Zeros
Decrease
Phase Advance
Decrease
These effects are larger when ck or dk are close the unit circle (i.e ∣ck∣,∣dk∣≈1).
All Pass Systems
Definition 17
All pass systems are those where ∣H(ejω)∣=k where k is some constant gain.
Their Z-transform has the real poles dk cancelled by real zeros, and complex poles ek cancelled by the conjugates ek⋆.
Theorem 5
If an All-Pass system is stable, then grd[H(ejω)]>0⟹ Causal and arg[H(ejω)]≤0⟹Phase Lag.
Minimum Phase Systems
Definition 18
A stable and causal system H(z) whose inverse H(z)1is also stable and causal is called a Minimum Phase System.
What this means is that all poles are zeros must be inside the unit circle, and the region of convergence is right sided. Minimum phase systems are called minimum phase because of all H(z) with the same magnitude response, a minimum phase system has the minimum phase and the minimum group delay.
Theorem 6
Any stable and causual system can be decomposed into a minimum phase system and an all-pass system.
H(z)=Hmin(z)Hap(z)
This is useful because if a signal undergoes a distortion, we can at least undo the minimum phase part of it (since Hmin has a guaranteed inverse).
Generalized Linear Phase Systems
Definition 19
A linear phase system is one with constant group delay.
H(ejω)=A(ejω)e−jαω⟹grd[H(ejω)]=α
Note that A(ejω) is a real function.
Definition 20
A generalized linear phase system has frequency response given by
H(ejω)=A(ejω)e−jαω+β⟹grd[H(ejω)]=α
If we limit ourselves to using FIR filters, then a GLP system must have either even or odd symmetry, meaning for some M
h[n]=h[M−n] or h[n]=−h[M−n].
This restricts us to 4 different filter types.
Symmetry
M
Filter Types
Notes
Type I
Even
Even
All
Type II
Even
Odd
Low Pass
Type III
Odd
Even
Bandpass
Type IV
Odd
Odd
High
Because of their symmetry, FIR systems are limited in where their zeros.
Type I, II: H(z)=z−MH(z−1)Type III, IV: H(z)=−z−MH(z−1)
In other words, if a=rejθ is a zero, then a⋆1 is too. We can decompose GLP systems into a minimum phase, maximum phase, and unit circle system.
Filter Design
The idea of filter design is to take a desired frequency response and design a filter which has that frequency response. Some frequency responses can only be described by IIR systems which are impractical for real applications, so we make various tradeoffs when we design FIR filters to implement in our systems. We also like our filters to be causal because it makes them usable in real-time systems.
A Mth order causal filter has M+1 coefficients.
Definition 21
The time-bandwidth product describes how sinc-like a filter looks like.
TBW=(M+1)πωc
The TBW is also the number of zero-crossings in the impulse response (including the zeros crossings at the end of the filter). To generate a High Pass filter, we can design a Low-Pass filter and then modulate it
hhp[n]=(−1)nhlp[n].
We can do the same for a bandpass filter
hbp[n]=2hlpcos(ω0n).
Windowing Method
One way to generate a filter which matches a desired frequency response is through windowing.
Choose a desired frequency response (often non-causal and IIR)
Window the Impulse Response
Module to shift the impulse response to make it casual
The length of the window impacts the transition width (how long it takes to transition). A longer window means a smaller width. The window type will impact the ripples in the frequency response. The choice of of window and its sidelobes impact these magnitudes.
Optimal Filter Design
With optimal filter design, we set up constraints to find a Hd(ejω) which approximates H(ejω) based on our optimization requirements. In general, we have some regions Wc⊆[0,π] that we care about and other regions that we don’t care about. We can first design a noncausal filter H(ejω)~ and then shift it to make it causal. We do this by sampling and discretizing the frequency response to ωk=kpπ where −π≤ω1≤...≤ωp≤π. We choose P to be sufficiently big and make sure the ωk∈ωc (the region we care about). In a least squares setup, we can solve
Other possible optimizations are Weightest Least Squares or Chebyshev Design.
WLS: min∫−ππW(ω)∣H(ejω)−Hd(ejω)∣2dω
Chebyshev: minw∈Wcmax∣H(ejω−Hd(ejω)∣
Another optimization technique is the min-max ripple design where we try and control the deviations of the filter from the desired response. We can set up a linear program to do this for us. For example, if we were designing a low pass filter, we could write the LP