π§ Transformperrins/class/F14_360/lab/...Β Β· 2014. 12. 2.Β Β· Unilateral z-Transform Region of...
Transcript of π§ Transformperrins/class/F14_360/lab/...Β Β· 2014. 12. 2.Β Β· Unilateral z-Transform Region of...
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π§ Transform Lab Notes 13
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Review Generalized CTFT -> Laplace Transform
π π = π₯ π‘ πβπ π‘ππ‘
β
ββ
π₯ π‘ =1
π2π π π π+π π‘ππ
π+πβ
πβπβ
π ππ = π₯ π‘ πβπππ‘ππ‘
β
ββ
π₯ π‘ =1
2π π ππ π+πππ‘ππ
β
ββ
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Fourier Family Continuous Frequency, π π Discrete Frequency, π π
Continuous Time, π₯ π‘
Discrete Time, π₯ π
Continuous Time Fourier Transform (CTFT)
Discrete Time Fourier Transform (DTFT)
Continuous Time Fourier Series (CTFS)
Discrete Time Fourier Series (DTFS) -OR- Discrete Fourier Transform (DFT)
Generalizes to Laplace Transform
Generalizes to π§ Transform
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New Idea Generalized DTFT -> π§ Transform
π π§ = π₯ π π§βπβ
π=ββ
π₯ π =1
π2π π π§ π§πβ1πz
πΆ
π ππΞ© = π₯ π πβπΞ©nβ
π=ββ
π₯[π] =1
2π π ππΞ© π+πΞ©nπΞ©2π
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Bilateral π§ Transform
π π§ = π₯ π π§βπβ
π=ββ
Region of Convergence for the bilateral z-Transform is dependent upon the nature of the discrete-time signal, π₯ π .
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Example π₯ π = 0.5 π π’ π
π₯ π = β¦ , 0.53, 0.52, 0.51, 1 , 0.51, 0.52, 0.53, β¦
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Find z Transform of x[n]
π π§ = 0.5 π π’ π π§βπβ
π=ββ
rewriting the sum since the amplitude values are the same,
π π§ = 2 0.5π π§βπβ
π=0
β 1
the sum is just the geometric series
0.5π π§βπβ
π=0
= 0.5
π§
πβ
π=0
=1
1 β0.5π§
=π§
π§ β 0.5, π§ > 0.5
β΄ π π§ =2π§
π§ β 0.5β 1 =
π§ + 0.5
π§ β 0.5, π§ > 0.5
ROC
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In the z-plane π π§ =π§ + 0.5
π§ β 0.5
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Poles and Zeros
Just like the Laplace transforms, z Transforms have poles and zeros also
The function in matlab to map these on the complex plane is called zplane().
This is similar to pzmap().
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Code
num1=[1 0.5];
den1=[1 -0.5];
zs1=roots(num1);
ps1=roots(den1);
zplane(zs1,ps1)
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Frequency Response
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Frequency Response
Additionally the function in matlab to show the frequency response of a discrete-time system is called freqz().
This is similar to freqs() for continuous-time systems.
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Code
num1=[1 0.5];
den1=[1 -0.5];
omega=linspace(-20,20,200);
freqz(num1,den1,omega)
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Multiplying polynomials
Question: How do I use MATLAB to multiply this expression?
π§2 + 2π§ + 10 π§ + 2
Answer: Use conv()!
conv([1 2 10],[1 2])
ans =
1 4 14 20
Note this function can be nested also!
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Unilateral z-Transform
Region of Convergence for the unilateral z-Transform is always the open exterior of a circle, centered at the origin of the z-plane whose radius is the largest finite pole magnitude.
Contrast this with the Region of Convergence for the unilateral Laplace Transform which is always the region of the s-plane to the right of all the finite poles.
π π§ = π₯ π π§βπβ
π=0