Signal Processing
Signal Processing Signal Processing
Transform theory The Fourier transform as the limit of the Fourier series Let x(t) be an aperiodic continuos-time signal with the property lim x(t) = 0 |t|→∞ Extend this signal as a periodic signal with period T so that ˜x(t) = ∞∑ n=−∞ x(t − nT ) lim ˜x(t) = x(t) for t ∈ [−T/2, T/2] T →∞ It follows that X k = 1 ∫ T /2 ˜x(t)e −i2π T k t dt → 1 ∫ T /2 x(t)e −i2π T k t dt → 1 T −T /2 T −T /2 T X( k T ) as T → ∞. Moreover, let f k = k/T → f and f k+1 − f k = 1/T → df, so that ˜x(t) = as T → ∞. ∞∑ k=−∞ X k e i2πf kt → ∞∑ k=−∞ ∫ 1 ∞ T X(f k)e i2πfkt → X(f)e i2πft df = x(t) −∞ Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 8(28)
Transform theory Classical Fourier transform: F : L 2 → L 2 ⎧ ∫ ∞ ⎪⎨ Fx(t) = x(t)e −iωt dt ⎪⎩ Important properties: ⎧ Duality: ⎪⎨ ⎪⎩ −∞ F −1 X(ω) = 1 2π Time-derivative: Frequency-derivative: Time-shift: ∫ ∞ −∞ X(ω)e iωt dω FFx(t) = 2πx(−t) F∂ t x(t) = iωX(ω) Ftx(t) = i∂ ω X(ω) Fx(t − t 0 ) = e −iωt0 X(ω) Frequency-shift: Fe iω0t x(t) = X(ω − ω 0 ) Scaling: Fx(at) = 1 |a| X(ω a ) Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 9(28)
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- Page 3 and 4: Transform theory ⎧ ⎪⎨ ⎪⎩
- Page 5 and 6: Transform theory Exercise 1 (b): Th
- Page 7: Transform theory Exercise 1 (d): Th
- Page 11 and 12: Transform theory The Parseval’s r
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- Page 15 and 16: Transform theory Convolution theore
- Page 17 and 18: Transform theory Convolution theore
- Page 19 and 20: Transform theory Convolution theore
- Page 21 and 22: Transform theory Convolution theore
- Page 23 and 24: Time-shift in the Fourier transform
- Page 25 and 26: Transform theory Time-derivative in
- Page 27 and 28: Transform theory Important symmetry
Transform theory<br />
Classical Fourier transform: F : L 2 → L 2<br />
⎧ ∫ ∞<br />
⎪⎨ Fx(t) = x(t)e −iωt dt<br />
⎪⎩<br />
Important properties:<br />
⎧<br />
Duality:<br />
⎪⎨<br />
⎪⎩<br />
−∞<br />
F −1 X(ω) = 1<br />
2π<br />
Time-derivative:<br />
Frequency-derivative:<br />
Time-shift:<br />
∫ ∞<br />
−∞<br />
X(ω)e iωt dω<br />
FFx(t) = 2πx(−t)<br />
F∂ t x(t) = iωX(ω)<br />
Ftx(t) = i∂ ω X(ω)<br />
Fx(t − t 0 ) = e −iωt0 X(ω)<br />
Frequency-shift: Fe iω0t x(t) = X(ω − ω 0 )<br />
Scaling: Fx(at) = 1<br />
|a| X(ω a )<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 9(28)
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