Signal Processing
Signal Processing Signal Processing
Transform theory Exercise 1 (a): The Fourier transform for continuous–time aperiodic signals { A 0 ≤ t ≤ Tp x(t) = 0 otherwise ∫ ∞ X(f) = x(t)e −i2πft dt = −∞ = A 1 − e−i2πfTp i2πf ∫ Tp [ e Ae −i2πft −i2πft ] Tp dt = A 0 −i2πf 0 = Ae −iπfTp eiπfTp − e −iπfTp i2πf −iπfTp sin(πfTp) = AT pe πfT p Zeros of X(f): X(f) = 0 ⇔ sin(πfT p) = 0 ⇔ πfT p = mπ ⇔ f = m T p , m = 0, ±1, ±2, . . . Draw a sketch of both the signal x(t) and its Fourier transform X(f). Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 4(28)
Transform theory Exercise 1 (b): The Fourier series for continuous–time T -periodic signals ⎧ A 0 ≤ t < T p ⎪⎨ ˜x(t) = 0 T p < t < T ⎪⎩ ˜x(t + T ) all t X k = 1 ∫ ˜x(t)e −i 2π T kt dt = 1 ∫ T ˜x(t)e −i 2π T kt dt = 1 ∫ Tp Ae −i 2π T kt dt T T T 0 T 0 { = same integral as above with f k = k } = 1 T T ATpe−iπ T k sin(π k Tp T Tp) π k T Tp Draw a sketch of both the signal ˜x(t) and its Fourier series X k . Note that the Fourier series coefficients X k of ˜x(t) are equal to 1 times the Fourier T transform of one single period x(t) of the periodic signal ˜x(t), sampled at the frequencies f = k T X k = 1 ∫ ˜x(t)e −i 2π T kt dt = 1 ∫ x(t)e −i 2π T kt dt = 1 ∫ ∞ x(t)e −i 2π T kt dt T T T T T −∞ = 1 T X(f)| f= k T Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 5(28)
- Page 1 and 2: Signal Processing Lecture 1 Sven No
- Page 3: Transform theory ⎧ ⎪⎨ ⎪⎩
- Page 7 and 8: Transform theory Exercise 1 (d): Th
- Page 9 and 10: Transform theory Classical Fourier
- Page 11 and 12: Transform theory The Parseval’s r
- Page 13 and 14: Transform theory The Parseval’s r
- 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 />
Exercise 1 (b): The Fourier series for continuous–time T -periodic signals<br />
⎧<br />
A 0 ≤ t < T p<br />
⎪⎨<br />
˜x(t) = 0 T p < t < T<br />
⎪⎩<br />
˜x(t + T ) all t<br />
X k = 1 ∫<br />
˜x(t)e −i 2π T kt dt = 1 ∫ T<br />
˜x(t)e −i 2π T kt dt = 1 ∫ Tp<br />
Ae −i 2π T kt dt<br />
T T<br />
T 0<br />
T 0<br />
{<br />
= same integral as above with f k = k }<br />
= 1 T T ATpe−iπ T k sin(π k Tp T Tp)<br />
π k T Tp<br />
Draw a sketch of both the signal ˜x(t) and its Fourier series X k .<br />
Note that the Fourier series coefficients X k of ˜x(t) are equal to 1 times the Fourier<br />
T<br />
transform of one single period x(t) of the periodic signal ˜x(t), sampled at the<br />
frequencies f = k T<br />
X k = 1 ∫<br />
˜x(t)e −i 2π T kt dt = 1 ∫<br />
x(t)e −i 2π T kt dt = 1 ∫ ∞<br />
x(t)e −i 2π T kt dt<br />
T T<br />
T T<br />
T −∞<br />
= 1 T X(f)| f= k T<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 5(28)
Change language