"EECE 365 Signals, Systems, & Transforms Ch 5. (Supp 04) DT Analysis Using Z-TransformMATLAB Session 1MATLAB Session • DT Frequency Domain Analysis z-Transform – Learn to use “linspace” function in MATLAB. – Find z-Transform from discrt. time signal using MATLAB function. – Find Inverse z-Transform from cont. frequency signal using MATLAB function. • Generating Amplitude Responses and Plots using MATLAB – Find z-Transform from discrt. time signal using MATLAB function and plot the result w.r.t. frequency (W-plane). • Design and Plot Digital Filters using MATLAB – LPF, HPF, BPF 2Finding z-Transform And Inverse z-Transform 3INLAB Report (No 1) :• Find z-Transform for following time domain signal using MATLAB. • Consider using “sym” function to define time domain signal. • Can you guess MATLAB function which performs z-Transform? • Use help in MATLAB to find the syntax for function. n? ? ? n x(n) ?a cos ? ? ? 2 ? ? 4INLAB Report (No 2) :• Consider a DT signal, x(n), and DT system, h(n), described as below. • Use z-Transform to find output response, Y(z), in z-domain using MATLAB. • Use inverse z-Transform to find DT output response, y(n). • Can you guess MATLAB function which performs z-Transform? • Use help in MATLAB to find the syntax for function. x(n) ?u(n) n h(n) ? (0.5) u(n) x(n) y(n) h(n) How did we find outputy(n)? 5[example] Discrete LTI system output x(n) ?u(n) n h(n) ? (0.5) u(n) x(n) y(n) h(n) How did we find outputy(n)? [Step 1] Convert into “Freq Domain” (z-Transform) 1 z X (z) ? ? ?1 x(n) ?u(n) 1?z z ?1 n 1 z h(n) ? (0.5) u(n) H (z) ? ? ?1 1? (0.5)z z ? 0.5[example] Discrete LTI system output x(n) ?u(n) n h(n) ? (0.5) u(n) x(n) y(n) h(n) How did we find outputy(n)? [Step 2] Use “Convolution Theorem” in z-Transform 1 z X (z) ? ? ?1 x(n) ?u(n) 1?z z ?1 n 1 z h(n) ? (0.5) u(n) H (z) ? ? ?1 1? (0.5)z z ? 0.5 y(n) ?x(n)*h(n) Y(z) ? X (z)H(z)[example] Discrete LTI system output x(n) ?u(n) n h(n) ? (0.5) u(n) x(n) y(n) h(n) How did we find outputy(n)? [Step 3] Use “Convolution Theorem” in z-Transform Y(z) ? X (z)H(z) z z ? ?? ? Y(z) ? ? ?? ? z ?1 z ? 0.5 ? ?? ?[example] Discrete LTI system output x(n) ?u(n) n h(n) ? (0.5) u(n) x(n) y(n) h(n) How did we find outputy(n)? [Step 4] Use “Partial Fraction Expansion” to find “y(n)” Y(z) ? X (z)H(z) z z ? ?? ? Y(z) ? ? ?? ? z ?1 z ? 0.5 ? ?? ? A = 2,B = -1 A B Y(z) ? ? z ?1 z ? 0.5[example] Discrete LTI system output x(n) ?u(n) n h(n) ? (0.5) u(n) x(n) y(n) h(n) How did we find outputy(n)? [Step 5] Use “(Inverse) z-Transform Pair” to find “y(n)” Y(z) ? X (z)H(z) z z ? ?? ? Y(z) ? ? ?? ? z ?1 z ? 0.5 ? ?? ? A = 2,B = -1 n n A B y(n) ? ?2(1) ? (?1)(0.5) ?u(n) Y(z) ? ? z ?1 z ? 0.5Generating Frequency Responses (Amplitude Response) 11DT Frequency Response Continuous Time Discrete Time • Amplitude & Phase Resp. • Amplitude & Phase Resp. (Bode Plot) s z 1 H(s) ? H(z) ? ? ?1 (s ?1)(s ?10) z ? 0.8 1? 0.8z -1 10 4.5 4 3.5 -2 3 10 2.5 2 1.5 1 -3 10 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 10 10 10 10 10 W Frequency (rad/s) 50 100 50 0 0 -50 -50 -100 -3 -2 -1 0 1 2 3 -2 -1 0 1 2 10 10 10 10 10 W Frequency (rad/s) Phase (degrees) Magnitude j W j WH[e ] [deg] |H[e ]| ?DT Frequency Response • To find the freq. response for a DT system – Find the transfer function of the DT system – ConvertjW assumed ? ? 0(on the unit circle) z ?e • This step is needed to find “amplitude” response and“phase” response of a DT system • (similar process as converting s-plane into freq plane ina CT system) ? j? s ?e eDT Frequency Response (cont.) • Terms being used – Analog frequency : ? Analog Frequency?T j?T z ?e e Related Term – Digital frequency : W DT converted freq jW z ?e Mapping – Relations W ? ?T Design -3 T : Sampling period ((ex) 10 [sec])assume,y[?1] ? y[?2] ? ... ? 0 ( ? eventually means find the “Transfer Function” of this system)assume,y[?1] ? y[?2] ? ... ? 0 ( ? eventually means find the “Transfer Function” of this system) y[n]?0.8y[n ?1] ?x[n] [Step 1] Convert into “Advanced Form” y[n ?1] y[n] y[n] y[n ?1] x[n ?1] x[n] “Delayed Form” “Advanced Form”assume,y[?1] ? y[?2] ? ... ? 0 ( ? eventually means find the “Transfer Function” of this system) y[n]?0.8y[n ?1] ?x[n] ?1 Y(z) ?0.8?Y(z)z ?y[?1]?? X (z) zero [Step 2] Perform z-Transform (Use Right Shift Properties) Y (z) y[n] u[n] ?1 Y (z)z ? y[?1] y[n ?1] u[n] X (z) x[n] u[n] Discrete Time Domain Freq Domain (z-plane)assume,y[?1] ? y[?2] ? ... ? 0 ( ? eventually means find the “Transfer Function” of this system) y[n]?0.8y[n ?1] ?x[n] ?1 Y(z) ?0.8?Y(z)z ?y[?1]?? X (z) zero Y(z) 1 H(z) ? ? ?1 X (z) (1? (0.8)z ) [Step 3] Find Transfer Function H(z)assume,y[?1] ? y[?2] ? ... ? 0 ( ? eventually means find the “Transfer Function” of this system) Y(z) 1 jW H(z) ? ? ?1 z ?e X (z) (1? (0.8)z ) [Step 4] System Response : “Amplitude Response” and “Phase Response” of H(z) jW z ?eINLAB Report (No 3) :• Generate Amplitude Plot and Phase Plot for following DT frequency response system using MATLAB. • Define a discrete-time frequency variable W from -? to -? with step size of ?/200. (Note, ? in MATLAB is pi) • Consider using “tf”, “freqresp”, and “squeeze” function to define frequency response of system. • Consider using “abs” function to plot amplitude response of system. • Consider using “angle” function to plot phase response of system. • Plot amplitude response w.r.t. DT frequency W . • Plot phase response w.r.t. DT frequency W . • Use help in MATLAB to find the syntax for function. • Include screen shot of your code and results in report. • Describe the result in your own words. 1 H(z) ? ?1 (1? (0.8)z ) 20[ Example 1 ] Poles and Zeros in Freq. Resp. (Graphical Representation)Poles and Zeros in a DT system • Similar to the CT system – Freq. Resp. of a DT system is determined by – the “location” of poles and zeros in the “transfer function” • Zeros ? increasing magnitude w.r.t. freq. • Poles ? decreasing magnitude w.r.t. freq. – In a CT system ? we introduced “imag” axis to analyze “amplitude” and “phase” – In a DT system ? we introduced “unit circle” to analyze “amplitude” and “phase”Poles and Zeros in a DT system • Consider a DT system with transfer function jW z ?e • We can rewrite this in polar form as()Poles and Zeros in a DT system jW • “Magnitude” of the DT system (at point ) e ?more zeros will increase magnitude ?more poles will decrease magnitude jW e • “Phase” of the DT system (at point ) ?more poles will ?more zeros will decrease phase increase phase? Considersame example again. assume,y[?1] ? y[?2] ? ... ? 0 ( ? eventually means find the “Transfer Function” of this system) Y(z) 1 H(z) ? ? ?1 X (z) (1? (0.8)z ) [Q] Considersame example again. How many “poles” and “zeros” are there in this system ???? Considersame example again. assume,y[?1] ? y[?2] ? ... ? 0 ( ? eventually means find the “Transfer Function” of this system) Y(z) 1 z (z ?z ) 1 H(z) ? ? ? ? ?1 X (z) (1? (0.8)z ) (z ? (0.8)) (z ?g ) 2 [Q] Considersame example again. One Zero atZ = 0 1 One Pole at g = 0.8 2? Considersame example again. assume,y[?1] ? y[?2] ? ... ? 0 ( ? eventually means find the “Transfer Function” of this system) Y(z) 1 z (z ?z ) 1 H(z) ? ? ? ? ?1 X (z) (1? (0.8)z ) (z ? (0.8)) (z ?g ) 2 jW complex polar notation z ?e jW j0 z ? 0 ? 0?e ? 0?e 1 One Zero atZ = 0 ? 1 1 jW j0 2 One Pole at g = 0.8 ? g ? 0.8 ? 0.8?e ? 0.8?e 2 2 [Q] Convert Transfer Function into “complex polar” representations.? Considersame example again. assume,y[?1] ? y[?2] ? ... ? 0 ( ? eventually means find the “Transfer Function” of this system) jW j0 Y(z) 1 z (1?e ? 0?e ) H(z) ? ? ? ? ?1 jW j0 X (z) (1? (0.8)z ) (z ? (0.8)) (1?e ? 0.8?e ) jW z ?1?e j0 g ? 0.8?e (pole) 2 Variable - Observation point jW z ?e (Unit circle, varies w.r.t. W) [Q] Now draw this on “complex polar plane ” j0 z ? 0?e (zero) 1Amplitude Response [Ex 5-10] When W=0 : jW j0 (1?e ? 0?e ) Y(z) 1 z r 1 H(z) ? ? ? ? ? ?1 jW j0 X (z) (1? (0.8)z ) (z ? (0.8)) d (1?e ? 0.8?e ) 2 : observation point r : distance from zero1 to observation point : distance from pole d 2 to observation point r 1 ? r d d 1 2 2 W=0W=0j0 j0 z ? 0?e (zero) g ? 0.8?e (pole) 1 2Amplitude Response [Ex 5-10] When W=?/4 : jW j0 (1?e ? 0?e ) Y(z) 1 z r 1 H(z) ? ? ? ? ? ?1 jW j0 X (z) (1? (0.8)z ) (z ? (0.8)) d (1?e ? 0.8?e ) 2 : observation point r : distance from zero1 to observation point : distance from pole d 2 to observation point W=?/4r r 1 1 ? d 2 d 2 W=?/4j0 j0 z ? 0?e (zero) g ? 0.8?e (pole) 1 2Amplitude Response [Ex 5-10] When W=2?/4 : jW j0 (1?e ? 0?e ) Y(z) 1 z r 1 H(z) ? ? ? ? ? ?1 jW j0 X (z) (1? (0.8)z ) (z ? (0.8)) d (1?e ? 0.8?e ) 2 : observation point r : distance from zero1 to observation point W=2?/4: distance from pole d 2 to observation point d 2 r r 1 1 ? d 2 W=2?/4j0 j0 z ? 0?e (zero) g ? 0.8?e (pole) 1 2Amplitude Response [Ex 5-10] When W=3?/4 : jW j0 (1?e ? 0?e ) Y(z) 1 z r 1 H(z) ? ? ? ? ? ?1 jW j0 X (z) (1? (0.8)z ) (z ? (0.8)) d (1?e ? 0.8?e ) 2 : observation point r : distance from zero1 to observation point W=3?/4: distance from pole d 2 to observation point d 2 r r 1 1 ? d 2 W=3?/4j0 j0 z ? 0?e (zero) g ? 0.8?e (pole) 1 2Amplitude Response [Ex 5-10] When W=4?/4 : jW j0 (1?e ? 0?e ) Y(z) 1 z r 1 H(z) ? ? ? ? ? ?1 jW j0 X (z) (1? (0.8)z ) (z ? (0.8)) d (1?e ? 0.8?e ) 2 : observation point r : distance from zero1 to observation point : distance from pole d 2 to observation point W=4?/4r d 1 2 ? d r 2 1 W=4?/4j0 j0 z ? 0?e (zero) g ? 0.8?e (pole) 1 2[ Example 2 ] continued Poles and Zeros in Freq. Resp. (Controlling Gains Using Poles and Zeros) Low Pass Filter MATLAB verificationsINLAB Report (No 4) :• We are trying to control gain (attenuation) using poles in DT frequency response. • Modify existing code to generate Amplitude Plot and Phase Plot for following DT frequency response system. • Plot amplitude response w.r.t. DT frequency W . • Plot phase response w.r.t. DT frequency W . • Use help in MATLAB to find the syntax for function. • Include screen shot of your code and results in report. • Describe the result in your own words. 1 1 (a) (c) H(z) ? H(z) ? z ? 0.8 z ? 0.5 1 1 (d) H(z) ? (b) H(z) ? z ? 0.3 z ? 0.7 35INLAB Report (No 5) :• We are trying to control gain (attenuation) adding more zeros in DT frequency response. • Modify existing code to generate Amplitude Plot and Phase Plot for following DT frequency response system. • Plot amplitude response w.r.t. DT frequency W . • Plot phase response w.r.t. DT frequency W . • Use help in MATLAB to find the syntax for function. • Include screen shot of your code and results in report. • Describe the result in your own words. z ? 0.8 1 (a) (c) H(z) ? H(z) ? z ? 0.8 z ? 0.8 z (b) H(z) ? z ? 0.8 36• [LPF]What Happens if you add zero(s) ??? 1 H(z) ? z ? 0.9 [Q] Modify your MATLAB code to see the results ??? z H(z) ? z ? 0.9 z ? 0.9 H(z) ? z ? 0.9• [LPF]What Happens if you add zero(s) ??? 1 H(z) ? z ? 0.9 z H(z) ? z ? 0.9 z ? 0.9 H(z) ? z ? 0.9• [LPF]What Happens if you add zero(s) ??? 1 – Enhances gain H(z) ? z ? 0.9 – But not as much as pole – Major effects on phase z H(z) ? z ? 0.9 z ? 0.9 H(z) ? z ? 0.9INLAB Report (No 6) :• We are trying to control gain (attenuation) adding more poles in DT frequency response. • Modify existing code to generate Amplitude Plot and Phase Plot for following DT frequency response system. • Plot amplitude response w.r.t. DT frequency W . • Plot phase response w.r.t. DT frequency W . • Use help in MATLAB to find the syntax for function. • Include screen shot of your code and results in report. • Describe the result in your own words. 1 (a) H(z) ? z ? 0.8 1 (b) H(z) ? (z ? 0.9)(z ? 0.8)(z ? 0.7) 40• [LPF]What Happens if you add multiple poles ??? 1 H(z) ? z ? 0.9 1 H(z) ? (z ? 0.9)(z ? 0.8)(z ? 0.7) 1 ? 3 2 z ? 2.4z ?1.91z ? 0.504• [LPF]What Happens if you add multiple poles ???[ Example 3 ] Poles and Zeros in Freq. Resp. (Controlling Gains Using Poles and Zeros) High Pass FilterINLAB Report (No 7) :• Generate Amplitude Plot and Phase Plot for following DT frequency response system using MATLAB. • Modify existing code to generate Amplitude Plot and Phase Plot for following DT frequency response system. • Plot amplitude response w.r.t. DT frequency W . • Plot phase response w.r.t. DT frequency W . • Use help in MATLAB to find the syntax for function. • Include screen shot of your code and results in report. • Describe the result in your own words. 1 1 (a) (c) H(z) ? H(z) ? z ? 0.5 z ? 0.8 1 1 (d) H(z) ? (b) H(z) ? z ? 0.3 z ? 0.7 44High Pass Filter (HPF) • How can you design DT HPF ??? • How can you describe magnitude of HPF ??? – Where is the peak (maximum gain / minimum loss) ?? – Where will you place pole(s) or zero(s) ???High Pass Filter (HPF) • How can you design DT HPF ?(Amplitude Response) ? You will need the min at W = 0 and peak at W = ? in Freq. Domain ? How can you achieve this trends in Freq. Domain ??High Pass Filter (HPF) • How can you design DT HPF ?(Amplitude Response) ? You will need the min at W = 0 and peak at W = ? in Freq. Domain ? How can you achieve this trends in Freq. Domain ?? ? What happens if you place a “pole” at W = ? in Freq. Domain ?? jW z ?e ? ?1 pole W?? 1 H(z) ? z ?1High Pass Filter (HPF) • How can you design DT HPF ??? 1 H(z) ? [Q] Modify your MATLAB code z ? 0.9 to see the results ??? 1 H(z) ? z ? 0.7 1 H(z) ? z ? 0.5 1 H(z) ? z ? 0.3• [HPF]What Happens if you add multiple poles ??? ? LPF (designed by multiple poles) ? HPF (designed by multiple poles)[ Example 4 ] Poles and Zeros in Freq. Resp. (Controlling Gains Using Poles and Zeros) Band Pass FilterINLAB Report (No 8) :• Generate Amplitude Plot and Phase Plot for following DT frequency response system using MATLAB. • Modify existing code to generate Amplitude Plot and Phase Plot for following DT frequency response system. • This time, consider using “freqz” function to define frequency response of system. • Plot amplitude response w.r.t. DT frequency W . • Plot phase response w.r.t. DT frequency W . • Use help in MATLAB to find the syntax for function. • Include screen shot of your code and results in report. • Describe the result in your own words. (a) When g = 0.99 (b) When g = 0.96 (z ?1)(z ?1) H(z) ? j? / 4 ?j? / 4 (c) When g = 0.83 (z? | g |e )(z? | g |e ) 2 2 2 (z ?1) (z ?1) (z ?1) ? ? ? 2 j? / 4 ?j? / 4 2 2 2 2 2 (z ? | g | (e ?e )z? | g | ) (z ? | g | (2cos(? / 4))z? | g | ) (z ? | g | ( 2)z? | g | ) 51Using MATLABUsing MATLAB ? For a DT system with below Transfer Function (z ?1)(z ?1) H(z) ? j? / 4 ?j? / 4 (z? | g |e )(z? | g |e ) ? Find amplitude and phase response of the system ? What happens to the amplitude responses when the location of poles vary?Using MATLAB ? Variation in the location of poles and frequency response (z ?1)(z ?1) H(z) ? j? / 4 ?j? / 4 (z? | g |e )(z? | g |e ) jW ? Convert into polar form( ) z ?e jW jW (e ?1)(e ?1) jW H(e ) ? jW j? / 4 jW ?j? / 4 (e ? | g |e )(e ? | g |e ) ? Magnitude becomes jW jW (e ?1)(e ?1) jW H(e ) ? jW j? / 4 jW ?j? / 4 (e ? | g |e ) (e ? | g |e )Using MATLAB ? Amplitude response of the system given by 2 jW jW * jW H(e ) ?H(e )H (e ) or 2 jW jW jW H(e ) ? H(e ) ? H(e ) ? Becomes the function of freq WUsing MATLAB ? Amplitude response w.r.t. freq W (z ?1)(z ?1) H(z) ? j? / 4 ?j? / 4 (z? | g |e )(z? | g |e )Using MATLAB ? Amplitude response w.r.t. freq W (z ?1)(z ?1) H(z) ? j? / 4 ?j? / 4 (z? | g |e )(z? | g |e )Using MATLAB ? Amplitude response w.r.t. freq W (z ?1)(z ?1) H(z) ? j? / 4 ?j? / 4 (z? | g |e )(z? | g |e )Using MATLAB ? Amplitude response w.r.t. freq W (z ?1)(z ?1) H(z) ? j? / 4 ?j? / 4 (z? | g |e )(z? | g |e )Using MATLAB ? Variation in the location of poles and frequency response (z ?1)(z ?1) H(z) ? j? / 4 ?j? / 4 (z? | g |e )(z? | g |e ) x x x x x x ? The peaking (resonance) becomes more pronounced as |g| approaches 1. ? Used in designing the filter shape"