Group Delay and Phase Delay (A) 7/9/2
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Phase Shift and Time Shift measure phase shift not in second but in portions of a cosine wave cycle within phase change in one cycle Phase Shift Delay in radians, degrees in seconds (time) Given time shift (delay) t sec The actual phase shift is different according to the frequency π,2π,3 π rad ω 2 ω 3 ω π rad 2 π rad 3 π rad ω 2 ω 3 ω t sec t sec The same delay applied to all frequencies The different phase shift to the different frequency Group Delay & Phase Delay 3 7/9/2
Frequency Response Frequency Response H (e j ω ) e j ω t LTI H (e j ω ) e jωt H (e j ω ) Magnitude Response LPF example H (e j ω ) Phase Response Group Delay & Phase Delay 4 7/9/2
Linear Phase System Linear Phase System Phase Shift Frequency H (e j ω ) ω Non-Linear Phase System Group Delay & Phase Delay 5 7/9/2
Uniform Time Delay () Frequency Response H (e j ω ) ω x(t) LTI y (t) 2 ω 3 ω The waveform shape can be preserved. uniform magnitude uniform time delay H (e j ω ) = c t sec π rad 2 π rad 3 π rad ω 2 ω 3 ω linear phase H (e j ω ) = k ω t sec Group Delay & Phase Delay 6 7/9/2
Uniform Time Delay (2) Frequency Response H (e j ω ) x(t) LTI y (t) Uniform Time Delay Could remove delay from the phase response to achieve a horizontal line at zero degree (No delay) The waveform shape can be preserved. uniform magnitude uniform time delay H (e j ω ) = c linear phase H (e j ω ) = k ω Group Delay & Phase Delay 7 7/9/2
CTFT of Sinc Function t = ±T, ±2T, ±3T, x(t) = x(t) T f s T X (f ) T T 2 T x(t) = sin(π t /T ) = sin(π f s t) π t /T π f s t t CTFT f s f s /2 + f s /2 H ( f ) = T, f f s /2, otherwise + f s f T X (f ) f s f s /2 + f s /2 + f s f T arg {X (f )} Real Symmetric Signal Zero Phase f s f s /2 + f s /2 + f s f Group Delay & Phase Delay 8 7/9/2
CTFT Time Shifting Property x(t) T f s T X (f ) T T 2 T x(t) = sin(π t /T ) = sin(π f s t) π t /T π f s t t CTFT f s f s /2 + f s /2 X (f ) = T, f f s /2, otherwise + f s f y (t) = x(t T ) Y (f ) = X (f ) e j2 π f T T T 2 T T t CTFT f s f s /2 + f s /2 e j 2π f T + f s f y (t) = x (t 2T ) Y (f ) = X (f ) e j2π f 2T T T 2 T T t CTFT f s f s /2 + f s /2 e j 2 π f 2T + f s f Group Delay & Phase Delay 9 7/9/2
CTFT of Sinc Function Shifted by T T T f s T e j 2π f T T 2 T x(t) = sin(π t /T ) = sin(π f s t) π t /T π f s t t CTFT f s f s /2 + f s /2 X (f ) = T, f f s /2, otherwise + f s f Arg Φ(f ) slope = d Φ d f Group Delay y (t) = x(t T ) Y (f ) = X (f ) e j2π f T = 2πT d Φ d ω = T d Φ d ω = T mag arg T f s f s /2 + f s /2 T + f s /2 + f s + f s f f s f s /2 f Pure Delay (No Dispersion) Linear Phase Change slope = 2π T Group Delay & Phase Delay 7/9/2
CTFT of Sinc Function Shifted by 2T T T f s T e j 2π f 2T T 2 T x(t) = sin(π t /T ) = sin(π f s t) π t /T π f s t t CTFT f s f s /2 + f s /2 X (f ) = T, f f s /2, otherwise + f s f Arg Φ(f ) slope = d Φ d f Group Delay y (t) = x (t 2T ) Y (f ) = X (f ) e j2π f 2T = 2π 2T d Φ d ω = 2T d Φ d ω = 2T mag arg T f s f s /2 + f s /2 T + f s /2 + f s + f s f f s f s /2 f Pure Delay (No Dispersion) Linear Phase Change slope = 2π 2T Group Delay & Phase Delay 7/9/2
Group Delay () Consider the cosine components at closely spaced frequencies and their phase shifts in relation to each other Group Delay: The phase shift changes for small changes in frequency small changes in frequency Δω LTI phase shift changes ΔΦ A uniform, waveform preserving phase response linear Constant Group Delay X Uniform Time Delay (linear phase) Group Delay & Phase Delay 2 7/9/2
Group Delay (2) Constant slope Constant Group Delay Linear Phase System Phase Shift Frequency H (e j ω ) No dispersion ω Non-Linear Phase System Dispersion Varying slope Varying Group Delay Group Delay & Phase Delay 3 7/9/2
Simple LPF Frequency Response Frequency Response R v i (t ) LTI v o (t ) v i (t ) C v o (t ) H ( j ω) = + j ω/ω A ( j ω) = H ( j ω) = ω = RC j ϕ( jω) H ( j ω) = A ( j ω)e +ω 2 /ω 2 ϕ( j ω) = arg{h ( j ω)} = tan ( ω/ω ) Magnitude Response H ( j ω) Phase Response arg {H ( j ω)}.8 ω = RC = ω =.5 RC =.5.6.4 -.5.2 - -6-4 -2 2 4 6 ω -.5-6 -4-2 2 4 6 ω Group Delay & Phase Delay 4 7/9/2
Simple LPF Group Delay Frequency Response R v i (t ) LTI v o (t ) v i (t ) C v o (t ) H ( j ω) = + j ω/ω H ( j ω) = A( j ω)e j ϕ( j ω) ω = RC τ (ω).8 ω = RC = A ( j ω) = +ω 2 /ω 2 ϕ( j ω) = tan ( ω/ω ) Group Delay.6.4.2 τ(ω) = d ϕ d ω = +ω 2 /ω 2 2 3 4 5 Frequency ω Group Delay & Phase Delay 5 7/9/2
Simple LPF Step Response Frequency Response R v i (t ) LTI v o (t ) v i (t ) C v o (t ) v o (t ) = e t τ ω = RC = τ Which is the group delay? τ (ω) ω =.8 RC =.8.6.4.2 2 3 4 5 6 Group Delay.6.4.2 Group delay is not constant Dispersion 2 3 4 5 Frequency ω Group Delay & Phase Delay 6 7/9/2
Simple LPF Narrow Band Signal Frequency Response R v i (t ) LTI v o (t ) v i (t ) C v o (t ) v o (t ) = e t τ ω = RC = τ τ (ω) When focusing a narrow band signal Output will be Time delayed by Amplitude scaled by τ (ω ) A (ω ) Group Delay.8.6.4.2 Narrow Band Constant A (ω ) Constant τ (ω ) Phase shifted by ϕ(ω ) 2 3 4 5 Frequency ω Group Delay & Phase Delay 7 7/9/2
Simple LPF Approximation () Magnitude Response A ( j ω) = H ( j ω) = / +ω 2 /ω 2 k A(ω k ).8.6 Constant Approximation A group of frequencies near Narrow Band ω k A (ω k ).4 [ω k B, ω k +B].2 B ω k -6-4 -2 2 4 6 ω k ω k Phase Response ϕ( j ω) = arg{h ( j ω)} = tan ( ω/ω ) k ϕ(ω k ).5.5 Linear Approximation A group of frequencies near ω k Narrow Band ϕ(ω k ) = ϕ(ω k ) + (ω ω k )ϕ' (ω k ) ω k -.5 - [ω k B, ω k +B] B ω k ω k -.5-6 -4-2 2 4 6 Group Delay & Phase Delay 8 7/9/2
Simple LPF Approximation (2) A (ω k ) = * ω k ω k ω k ω k ϕ(ω k ) ω k ω k = * ω k ω k 2 A (ω k ) B sin [B(t+ϕ'(ω k))] π B(t+ϕ' (ω k )) A (ω k ) else ( B ω +B) π[e + jϕ( ω k) δ(ω ω k ) + e j ϕ(ω k) δ(ω+ω k )] Group Delay & Phase Delay 9 7/9/2
Simple LPF Approximation (3) A (ω k ) ω k ω k = slope = ϕ' (ω k ) * time shifting ω k ω k ϕ(ω k ) ω k ω k = * ω k ω k Phase = ϕ(ω k ) Frequency Domain A (ω k ) else ( B ω +B) π[e + jϕ( ω k) δ(ω ω k ) + e j ϕ(ω k) δ(ω+ω k )] Group Delay * Phase Delay Time Domain 2 A (ω k ) B sin [B(t+ϕ'(ω k))] π B(t+ϕ' (ω k )) x cos(ω k t+ϕ(ω k )) Group Delay & Phase Delay 2 7/9/2
Beat Signal Very similar frequency signals. Hz.9 Hz cos(2 π. t) cos(2 π.9 t) cos(2 π. t) + cos(2 π.9 t) = cos(2 π (..9) 2 t) cos(2 π (.+.9) 2 t) = cos(2 π. t) cos(2 π. t) Slow moving envelop Fast moving carrier Bandlimited Narrowband Signal 2.5 2.5.5 -.5 - -.5-2 -2.5 2.5 2.5.5 -.5 - -.5-2 -2.5 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 Group Delay & Phase Delay 2 7/9/2
Example RC Circuit & Frequency Response Frequency Response R v i (t ) LTI v o (t ) v i (t ) C v o (t ) H ( j ω) = + j ω/ω ω = H ( j ω) = +ω 2 arg {H ( j ω)} = tan ( ω) arg {H ( j ω)} = tan ( ω) Magnitude Response H ( j ω) Phase Response arg {H ( j ω)}.8.6 H ( j.9) =.743 H ( j.) =.77 H ( j.) =.672.5.5 H ( j.9) =.73 H ( j.) =.79 H ( j.) =.83.4 -.5.2 - -6-4 -2 2 4 6 ω -.5-6 -4-2 2 4 6 ω Group Delay & Phase Delay 22 7/9/2
Example Some Signal Responses Frequency Response cos(.9t) cos(.t) cos(.t) LTI LTI LTI.743 cos(.9t.73).77 cos(.t.79).672 cos(.t.83) H ( j.9) =.743 H ( j.) =.77 H ( j.) =.672 H ( j.9) =.73 H ( j.) =.79 H ( j.) =.83.5.5 -.5 -.5-5 5 2 25 3 35 4 sec - 5 5 2 25 3 35 4 sec Group Delay & Phase Delay 23 7/9/2
Example Beat Signal Response Frequency Response cos(.9t) + cos(.t) = 2cos(.t)cos(.t) LTI.743 cos(.9t.73) +.672 cos(.t.83) 2 2.5.5.5.5 -.5 -.5 - - -.5-2 2 3 4 5 6 7 8 sec -.5-2 2 3 4 5 6 7 8 sec Group Delay & Phase Delay 24 7/9/2
Example Envelopes Frequency Response τ(ω) = d ϕ d ω = +ω 2 τ() = 2 cos(.9t) + cos(.t) = 2cos(.t)cos(.t) LTI.743 cos(.9t.73) +.672 cos(.t.83).4cos(.(t.5))cos(.t π/ 4) 2 2.5.5.5.5 -.5 -.5 - - -.5-2 2 3 4 5 6 7 8 sec -.5-2 2 3 4 5 6 7 8 sec Group Delay & Phase Delay 25 7/9/2
Example Group & Phase Delay τ (ω) = d ϕ d ω = +ω 2 2.5 τ () = 2 t g =.5 arg {H ()} = tan ().5 t p = π/ 4 =.78 2.5 2.5.5 -.5.5 -.5 -.5 - - - -.5-2 2 3 4 5 6 7 8 sec Group Delay & Phase Delay 26 -.5-2 2 3 4 5 6 7 8-2 sec 7/9/2
Phase & Group Delay from Phase Response Phase Response Phase Delay t p t p = tan () = π/4 =.785 Phase Response ω Group Delay slope = t g t g =.5 ω Group Delay & Phase Delay 27 7/9/2
Simple LPF Step Response Frequency Response R v i (t ) LTI v o (t ) v i (t ) C v o (t ) v o (t ) = e t τ Which is the group delay? ω = RC = τ Group delay with a narrow band signal.8.6.4.2 2 3 4 5 6 Group delay is not constant Dispersion Bandlimited Narrow band Group delay Group Delay & Phase Delay 28 7/9/2
Angle and Angular Speed distance Φ Slope = time Φ = ω t t = Φ ω ω speed distance ΔΦ ΔΦ = Δω Δt Δt = ΔΦ Δω ω ω 2 Δω speed Group Delay & Phase Delay 29 7/9/2
Group Delay & Phase Delay 3 7/9/2
References [] http://en.wikipedia.org/ [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 23 [3] http://www.libinst.com/tpfd.htm [4] K.Shin, J.K. Hammond, Fundamentals of Signal Processing for Sound and Vibration Engineers [5] www.radiolab.com.au/designfile/dn4.pdf 7/9/2