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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
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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
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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
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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
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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
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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
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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
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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
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FR example MA filter n -1 1 2 3 4 5 6 Dan Ellis
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FR example MA filter: Response to ...
(jumps at sign changes: r=Mw/2p) Dan Ellis
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FR example MA filter input w0 = 0.1p H(ejw0) ~ 4/5 ejf0 w1 = 0.5p H(ejw1) ~ -1/5 ejf1 output Dan Ellis
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2. Transfer Function (TF)
Linking LCCDE, ZT & Freq. Resp... LCCDE: Take ZT: Hence: or: Transfer function H(z) Dan Ellis
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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
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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
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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
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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
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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
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TF FR zi, li are TF roots on z-plane Magnitude response
Phase response Dan Ellis
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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
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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
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FR: Geometric Interpretation
Magnitude and phase of a single zero: Pole is reciprocal mag. / negated phase Dan Ellis
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FR: Geometric Interpretation
Multiple poles / zeros: M Dan Ellis
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Geom. Interp. vs. 3D surface
3D magnitude surface for same system Dan Ellis
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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
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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
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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
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Filtering example Filter IR Freq. resp input/ output Dan Ellis
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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
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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
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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
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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 wcwm but q(wc-wm) = ql ; q(wc+wm) = qu ... Dan Ellis
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phase shift of envelope
Group Delay phase shift of carrier phase shift of envelope Dan Ellis
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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
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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
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