1. Frequency Response (FR) Transfer Function (TF)
ELEN E4810: Digital Signal Processing Topic 5: Transform-Domain Systems
Frequency Response (FR)
Transfer Function (TF)
Phase Delay and Group Delay
Dan Ellis

2. 1. Frequency Response (FR)
Fourier analysis expresses any signal as the sum of sinusoids e.g. IDTFT:
Sinusoids are the eigenfunctions of LSI systems (only scaled, not ‘changed’)
Knowing the scaling for every sinusoid fully describes system behavior
 frequency response
describes how a system affects each pure frequency
Dan Ellis

3. Sinusoids as Eigenfunctions
IR h[n] completely describes LSI system:
Complex sinusoid input i.e. 
Output is sinusoid scaled by FT at w0
x[n]
h[n]
y[n] = x[n] h[n] *
H(ejw) = |H(ejw)|ejq(w)
Dan Ellis

4. System Response from H(ejw)
If x[n] is a complex sinusoid at w0 then the output of a system with IR h[n] is the same sinusoid scaled by |H(ejw)| and phase-shifted by arg{H(ejw)} = q(w) where H(ejw) = DTFT{h[n]}
(Any signal can be expressed as sines...)
|H(ejw)| “magnitude response”  gain
arg{H(ejw)} “phase resp.”  phase shift
Dan Ellis

5. Real Sinusoids In practice signals are real e.g.
Real h[n]  H(e-jw) = H*(ejw) = |H(ejw)|e-jq(w) w
- w0 w0
X(ejw)
Dan Ellis

6. Real Sinusoids
A real sinusoid of frequency w0 passed through an LSI system with a real impulse response h[n] has its gain modified by |H(ejw0)| and its phase shifted by q(w0).
Acos(w0n+f)
h[n]
|H(ejw0)|Acos(w0n+f+q(w0))
Dan Ellis

7. Transient / Steady State
Most signals start at a finite time e.g. What is the effect?
Steady state - same as with pure sine input
Transient response - consequence of gating
Dan Ellis

8. Transient / Steady State
FT of IR h[n]’s tail from time n onwards
zero for FIR h[n] for n ≥ N
tends to zero with large n for any ‘stable’ IR
transient
Dan Ellis

9. FR example
MA filter  n -1 1 2 3 4 5 6
Dan Ellis

10. FR example MA filter: Response to ...
(jumps at sign changes: r=Mw/2p)
Dan Ellis

11. FR example
MA filter
input w0 = 0.1p  H(ejw0) ~ 4/5 ejf0 w1 = 0.5p  H(ejw1) ~ -1/5 ejf1
output
Dan Ellis

12. 2. Transfer Function (TF)
Linking LCCDE, ZT & Freq. Resp...
LCCDE:
Take ZT:
Hence:
or:
Transfer function H(z)
Dan Ellis

13. Transfer Function (TF)
Alternatively,
ZT 
Note: same H(z)
e.g. FIR filter, h[n] = {h0, h1,... hM-1}
 pk=hk, d0=1, DE is
... if system has DE form
... from IR
Dan Ellis

14. Transfer Function (TF)
Hence, MA filter:
Im{z}
zM=1 i.e. M roots of z=ejrp/M
Re{z}  1
ROC?
z=1 cancels
z-plane
Dan Ellis

15. TF example
y[n] = x[n-1] x[n-2] + x[n-3] y[n-1] y[n-2] y[n-3]
factorize:
z0 = 0.6+j0.8 l0 = 0.3 l1 = 0.5+j0.7 
Dan Ellis

16. TF example Poles li  ROC causal  ROC is |z| > max|li|
Im{z} z0 l1 
Re{z} l0  1
l*1
Poles li  ROC
causal  ROC is |z| > max|li|
includes u.circle  stable 
z*0
Dan Ellis

17. TF  FR DTFT H(ejw) = ZT H(z)|z = ejw
i.e. Frequency Response is Transfer Function eval’d on Unit Circle
factor:
Dan Ellis

18. TF  FR zi, li are TF roots on z-plane Magnitude response
Phase response
Dan Ellis

19. FR: Geometric Interpretation
Have
On z-plane:
Constant/ linear part
Product/ratio of terms related to poles/zeros
Each (ejw - n) term corresponds to a vector from pole/zero n to point ejw on the unit circle
Im{z} 
ejw-li
ejw-zi
Re{z}
ejw
Overall FR is product/ratio of all these vectors
Dan Ellis

20. FR: Geometric Interpretation
Magnitude |H(ejw)| is product of lengths of vectors from zeros divided by product of lengths of vectors from poles
Phase q(w) is sum of angles of vectors from zeros minus sum of angles of vectors from poles
Im{z} 
ejw-li
ejw-zi
Re{z}
ejw zi li
Dan Ellis

21. FR: Geometric Interpretation
Magnitude and phase of a single zero:
Pole is reciprocal mag. / negated phase
Dan Ellis

22. FR: Geometric Interpretation
Multiple poles / zeros: M
Dan Ellis

23. Geom. Interp. vs. 3D surface
3D magnitude surface for same system
Dan Ellis

24. Geom. Interp: Observations
Roots near unit circle  rapid changes in magnitude & phase
zeros cause mag. minima (= 0  on u.c.)
poles cause mag. peaks ( 1÷0= at u.c.)
rapid change in relative angle  phase
Pole and zero ‘near’ each other cancel out when seen from ‘afar’; affect behavior when z = ejw gets ‘close’
Dan Ellis

25. Filtering
Idea: Separate information in frequency with constructed H(ejw)
e.g.
Construct a filter: |H(ejw1)| ~ 1 |H(ejw2)| ~ 0
Then
interested in this part
don’t care about this part w w2 w1
-w1
-w2
H(ejw)
X(ejw)
“filtered out”
Dan Ellis

26. Filtering example 3 pt FIR filter h[n] = {a b a}-1 1 2 3 4 -2 a b
3 pt FIR filter h[n] = {a b a}
mix of w1 = 0.1 rad/samp and w2 = 0.4 rad/samp
 b = 13.46, a =
 to remove i.e. make H(ejw1) = 0
want = 1 at w = = 0 at w = M
Dan Ellis

27. Filtering example Filter IR Freq. resp input/ output Dan Ellis

28. 3. Phase- and group-delay
For sinusoidal input x[n] = cosw0n, we saw
i.e. or
where is phase delay
gain
phase shift or time shift
subtraction so positive tp means delay (causal)
Dan Ellis

29. Phase delay example For our 3pt filter:n -1 1 2 3 4 -2 a b
For our 3pt filter:
i.e. 1 sample delay (at all frequencies) (as observed)
Dan Ellis

30. Group Delay
Consider a modulated carrier e.g. x[n] = A[n]·cos(wcn) with A[n] = Acos(wmn) and wm << wc
Delay of envelope and carrier may differ
Dan Ellis

31. Group Delay So: x[n] = Acos(wmn)·cos(wcn) Now:
X(ejw)
wc +wm
wc -wm
So: x[n] = Acos(wmn)·cos(wcn)
Now:
Assume |H(ejw)| ~ 1 around wcwm but q(wc-wm) = ql ; q(wc+wm) = qu ...
Dan Ellis

32. phase shift of envelope
Group Delay
phase shift of carrier
phase shift of envelope
Dan Ellis

33. Group Delay If q(wc) is locally linear i.e. q(wc+Dw) = q(wc) + SDw,
Then carrier phase shift so carrier delay , phase delay
Envelope phase shift
 delay group delay
Dan Ellis

34. Group Delay
If q(w) is not linear around wc, A[n] suffers “phase distortion”  correction... n w wc
tp, phase delay
tg, group delay
q(w)
Dan Ellis