ELEN E4810: Digital Signal Processing Topic 5: Transform-Domain Systems

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

1. Frequency Response (FR) • Fourier analysis expresses any signal as the sum of sinusoidse.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 eachpure frequency Dan Ellis

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

System Response from H(ejw) • If x[n] is a complex sinusoid at w0then 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

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

h[n] Acos(w0n+f) |H(ejw0)|Acos(w0n+f+q(w0)) Real Sinusoids • A real sinusoid of frequency w0passed 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). Dan Ellis

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

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

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

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

FR example • MA filter • inputw0 = 0.1pH(ejw0) ~ 4/5 ejf0w1 = 0.5pH(ejw1) ~ -1/5 ejf1 • output Dan Ellis

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

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

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

TF example • y[n] = x[n-1] - 1.2 x[n-2] + x[n-3] + 1.3 y[n-1] - 1.04 y[n-2] + 0.222 y[n-3] • factorize: z0 = 0.6+j0.8l0 = 0.3l1 = 0.5+j0.7 Dan Ellis

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

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

TF  FR zi, li are TF roots on z-plane Magnituderesponse Phaseresponse Dan Ellis

Im{z} ejw-zi  ejw ejw-li  Re{z}  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 Overall FR is product/ratio ofall these vectors Dan Ellis

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

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

FR: Geometric Interpretation • Multiple poles / zeros: M Dan Ellis

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

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

X(ejw) H(ejw) w -w2 -w1 w1 w2 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 aboutthis part “filteredout” Dan Ellis

b a a n -2 -1 1 2 3 4 Filtering example • 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 = -6.76...  to removei.e. makeH(ejw1) = 0 want = 1 at w = 0.4 = 0 at w = 0.1 ... M Dan Ellis

Filtering example • FilterIR • Freq.resp • input/output Dan Ellis

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

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

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

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

Group Delay phase shiftof carrier phase shiftof envelope Dan Ellis

Group Delay • If q(wc)is locally linear i.e. q(wc+Dw) = q(wc) + SDw, • Then carrierphase shiftso carrier delay , phase delay • Envelopephase shift  delay group delay Dan Ellis

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

ELEN E4810: Digital Signal Processing Topic 5: Transform-Domain Systems