SLIDE 1
6.011: Signals, Systems & Inference
Lec 3 Energy spectral density
1
−
← ←
−
Inner (dot) product of signals
X < x, v >= x[k]v[k] for real signals x[·], v[·]
k
More generally, p[n] = X
k
x[k]v[k − n] = x∗v
←[n]
where v
←[`] = v[−`]
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6.011: Signals, Systems & Inference Lec 3 Energy spectral - - PowerPoint PPT Presentation
6.011: Signals, Systems & Inference Lec 3 Energy spectral - - PowerPoint PPT Presentation
6.011: Signals, Systems & Inference Lec 3 Energy spectral density 1 Inner (dot) product of signals X < x , v > = x [ k ] v [ k ] for real signals x [ ] , v [ ] k [ n ] X More generally, p [ n ] = x [ k ] v [ k n ]
SLIDE 2
SLIDE 3
− ← − −
Transform of reversed signal
DTFT of v[n] = V (ejΩ) = X
n
v[n]e−jΩn DTFT of v
←[n] =
X
n
v[−n]e−jΩn = X
k
v[k]ejΩk = V ∗(ejΩ) = V (e−jΩ)
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SLIDE 4
← −
- ←
−
Transform of inner product
p[n] = X
k
l DTFT of v
←[n] = V (e−jΩ) so
x[k]v[k n] = x⇤v
←[n]
P(ejΩ) = X(ejΩ)V (e−jΩ)
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SLIDE 5
Zero lag (n=0): Parseval, Rayleigh, Plancherel
π
1 Z p[0] = X x[k]v[k] = P (ejΩ)dΩ 2π
−π k
so 1 Z π
jΩ)V (e −jΩ)dΩ
X x[k]v[k] = X(e 2π
−π k
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SLIDE 6
− − −
Energy
Setting v[k] = x[k] , 1 Z π Ex = X |x[k]|2 = 2π
π k
1 Z π = |X(ejΩ)|2dΩ 2π
π
X(ejΩ)X(e−jΩ)dΩ
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SLIDE 7
r[n] =
- Energy spectral density (ESD)
X ¯ x[k]x[k n] = Rxx[n]
k
| {z } deterministic autocorrelation l transform pair
2
X(ejΩ) = S ¯
xx(ejΩ)
| {z } energy spectral density
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SLIDE 8
- ←
− −
Cross (energy) spectral density
¯ Ryx[n] = X
k
S ¯
yx(ejΩ) = Y (ejΩ)X(e–jΩ)
So if y[n] = h ⇤ x[n] then S ¯
yx(ejΩ) = H(ejΩ)X(ejΩ)X(e–jΩ) = H(ejΩ)S
¯
xx(ejΩ)
y[k]x[k n] = y⇤
←
x[n] l
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SLIDE 9
Similarly …
... if y[n] = h ∗ x[n] then
−jΩ)
S ¯
yy(ejΩ) =
Y (ejΩ)Y (e
−jΩ)
= H(ejΩ)X(ejΩ)X(e
−jΩ)H(e 2 jΩ)
¯
jΩ)
=
- H(e
- Sxx(e
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SLIDE 10
Energy of x[.] in a specified band
x[n] H(ejÆ) H(ejÆ) y[n] 1 ¢ ¢
- Æ0
Æ0 Æ
10
SLIDE 11
Noise-free signal
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SLIDE 12
Magnitude spectrum of noise-free signal
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SLIDE 13
ESD of noise-free signal
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SLIDE 14
+ unit-intensity white Gaussian noise
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SLIDE 15
Magnitude spectrum of noisy signal
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SLIDE 16
ESD of noisy signal
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SLIDE 17
Heart rate variability
ECG signal (a) −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 ECG amplitude (mV) 54 56 58 60 62 64 66 Time (sec) Instantaneous HR signal 45 50 55 60 65 70 75 80 Time (sec) (b) 1.1 1.15 1.2 1.25 1.3 x(t) (beats/sec) Power spectrum (c) 10 20 30 40 50 60 70 80 90 D
xx (e ) jÆ
−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 Frequency (Hz) 1
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6.011 Signals, Systems and Inference
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