SLIDE 1
6.011: Signals, Systems & Inference
Lec 2 Transforms
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−
DT convolution to z-transform (and system function)
∞
y[n] = X h[k]x[n − k]
k= −∞ n
x[n] = z , all n
∞
⇣ X
k⌘ n
y[n] = h[k]z z
k= −∞
| {z }
H(z0)
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6.011: Signals, Systems & Inference Lec 2 Transforms 1 DT - - PowerPoint PPT Presentation
6.011: Signals, Systems & Inference Lec 2 Transforms 1 DT - - PowerPoint PPT Presentation
6.011: Signals, Systems & Inference Lec 2 Transforms 1 DT convolution to z-transform (and system function) X y [ n ] = h [ k ] x [ n k ] k = n x [ n ] = z all n , 0 X k n y [ n ] = h [ k ] z z 0 0 k =
SLIDE 2
SLIDE 3
− −
−
DT convolution to DTFT (and frequency response)
∞
y[n] = X h[k]x[n k]
k= −∞ n jΩ0 )
x[n] = (e , all n, and π < Ω0 ≤ π
∞
⇣ X
jΩ0k⌘ jΩ0 )n
y[n] = h[k]e (e
k= −∞
| {z }
H(ejΩ0 )
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SLIDE 4
Using frequency response to specify response to sinusoidal inputs
x[n] = A cos(Ω0n + θ) y[n] = |H(ejΩ0 )| A cos ⇣ Ω0n + θ + ∠H(ejΩ0 ) ⌘
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SLIDE 5
− −
Frequency response (DTFT of unit sample response)
∞ jΩk
H(ejΩ) = X h[k]e
k= −∞
When is the above infinite sum well-defined? 1 Z π h[n] = H(ejΩ)ejΩn dΩ 2π
π
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SLIDE 6
− −
Spectral content of a signal (DTFT of the signal)
∞
X
jΩk
X(ejΩ) = x[k]e
k= −∞
1 Z π x[n] = X(ejΩ)ejΩn dΩ 2π
π
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SLIDE 7
Classes of DT signals that have DTFTs
Absolutely summable (finite “action”, have continuous spectra) – or not, but … Square summable (finite “energy”, spectra have discontinuities) – or not, but … Bounded (finite amplitude, spectra involve generalized functions like impulses)
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SLIDE 8
Convolution in time to multiplication in frequency
Putting together frequency response, spectral content, and superposition, we find
y[n] = h ∗ x[n]
in the time domain translates to
Y (ejΩ) = H(ejΩ)X(ejΩ)
in the frequency domain.
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SLIDE 9
Exercise
Derive the DTFT of
∞
r[n] = X x[k]x[k − n]
k= −∞
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MIT OpenCourseWare https://ocw.mit.edu
6.011 Signals, Systems and Inference
Spring 2018 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms.
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