7: Fourier Transforms: Convolution and Parseval s Theorem
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- Rudolf Hall
- 7 years ago
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1 Convolution Parseval s E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
2 Multiplication of Signals Question: What is the Fourier transform ofw(t) = u(t)v(t)? Convolution Parseval s E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
3 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
4 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
5 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) [Note use of different dummy variables] = + h= U(h)ei2πht dh + g= V(g)ei2πgt dg E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
6 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) [Note use of different dummy variables] = + h= U(h)ei2πht dh + g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
7 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) [Note use of different dummy variables] = + h= U(h)ei2πht dh + g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + f= V(f h)ei2πft df dh E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
![U(h)ei2πht dh + g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the](/docs-images/48/23984661/images/page_7.jpg "second integral: g = f h = + h= U(h) + f= V(f h)ei2πft df dh E.")
8 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) [Note use of different dummy variables] = + h= U(h)ei2πht dh + g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
![g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = +](/docs-images/48/23984661/images/page_8.jpg "h= U(h) + = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] E.")
9 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) [Note use of different dummy variables] = + h= U(h)ei2πht dh + g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] = + f= W(f)ei2πft df E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
![V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h)](/docs-images/48/23984661/images/page_9.jpg "+ = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] = + f= W(f)ei2πft df E.")
10 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) [Note use of different dummy variables] = + h= U(h)ei2πht dh + g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] = + f= W(f)ei2πft df wherew(f) = + U(h)V(f h)dh h= E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
![V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f=](/docs-images/48/23984661/images/page_10.jpg "f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] = + f= W(f)ei2πft df wherew(f) = + U(h)V(f h)dh h= E.")
11 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) [Note use of different dummy variables] = + h= U(h)ei2πht dh + g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] = + f= W(f)ei2πft df wherew(f) = + U(h)V(f h)dh U(f) V(f) h= E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
![dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f= f= V(f](/docs-images/48/23984661/images/page_11.jpg "h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] = + f= W(f)ei2πft df wherew(f) = + U(h)V(f h)dh U(f) V(f) h= E.")
12 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) [Note use of different dummy variables] = + h= U(h)ei2πht dh + g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] = + f= W(f)ei2πft df wherew(f) = + h= U(h)V(f h)dh U(f) V(f) This is the convolution of the two spectrau(f) andv(f). E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
![+ g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft](/docs-images/48/23984661/images/page_12.jpg "dhdf [swap ] = + f= W(f)ei2πft df wherew(f) = + h= U(h)V(f h)dh U(f) V(f) This is the convolution of the two spectrau(f) andv(f). E.")
13 Multiplication of Signals Convolution Parseval s Question: What is the Fourier transform ofw(t) = u(t)v(t)? Letu(t) = + h= U(h)ei2πht dh and v(t) = + g= V(g)ei2πgt dg w(t) = u(t)v(t) [Note use of different dummy variables] = + h= U(h)ei2πht dh + g= V(g)ei2πgt dg = + h= U(h) + g= V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ] = + f= W(f)ei2πft df wherew(f) = + U(h)V(f h)dh U(f) V(f) h= This is the convolution of the two spectrau(f) andv(f). w(t) = u(t)v(t) W(f) = U(f) V(f) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 2 /
![V(g)ei2π(h+g)t dgdh [mergee ( ) ] Now we make a change of variable in the second integral: g = f h = + h= U(h) + = f= f= V(f h)ei2πft df dh + h= U(h)V(f h)ei2πft dhdf [swap ]](/docs-images/48/23984661/images/page_13.jpg "= + f= W(f)ei2πft df wherew(f) = + U(h)V(f h)dh U(f) V(f) h= This is the convolution of the two spectrau(f) andv(f). w(t) = u(t)v(t) W(f) = U(f) V(f) E.")
14 Multiplication Example Convolution Parseval s u(t) = e at t a= t < Time (s) u(t) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
15 Multiplication Example u(t) = e at t t < u(t) a= Time (s) Convolution U(f) = a+i2πf [from before] Parseval s E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
![before] Parseval s E.](/docs-images/48/23984661/images/page_15.jpg "Fourier Series and Transforms (24-5559) Fourier")
16 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < [from before] u(t) U(f) a= Time (s) Frequency (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
17 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < [from before] u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
18 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < v(t) = cos2πft [from before] u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) v(t) Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
19 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < v(t) = cos2πft [from before] V(f) =.5(δ(f +F)+δ(f F)) u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) v(t) Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
20 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < v(t) = cos2πft [from before] V(f) =.5(δ(f +F)+δ(f F)) u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) v(t) V(f) Time (s).4 F= Frequency (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
21 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < v(t) = cos2πft [from before] V(f) =.5(δ(f +F)+δ(f F)) w(t) = u(t)v(t) u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) v(t) V(f) w(t) Time (s).4 F= Frequency (Hz) a=2, F=2-5 5 Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
22 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < v(t) = cos2πft [from before] V(f) =.5(δ(f +F)+δ(f F)) w(t) = u(t)v(t) W(f) = U(f) V(f) u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) v(t) V(f) w(t) Time (s).4 F= Frequency (Hz) a=2, F=2-5 5 Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
23 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < v(t) = cos2πft [from before] V(f) =.5(δ(f +F)+δ(f F)) w(t) = u(t)v(t) W(f) = U(f) V(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) v(t) V(f) w(t) Time (s).4 F= Frequency (Hz) a=2, F=2-5 5 Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
24 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < v(t) = cos2πft [from before] V(f) =.5(δ(f +F)+δ(f F)) w(t) = u(t)v(t) W(f) = U(f) V(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) v(t) V(f) w(t) W(f) Time (s).4 F= Frequency (Hz) a=2, F=2-5 5 Time (s) Frequency (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
25 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < v(t) = cos2πft [from before] V(f) =.5(δ(f +F)+δ(f F)) w(t) = u(t)v(t) W(f) = U(f) V(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) v(t) V(f) w(t) W(f) <W(f) (rad/pi Time (s).4 F= Frequency (Hz) a=2, F=2-5 5 Time (s) Frequency (Hz) Frequence (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
26 Multiplication Example Convolution Parseval s u(t) = U(f) = a+i2πf e at t t < v(t) = cos2πft [from before] V(f) =.5(δ(f +F)+δ(f F)) w(t) = u(t)v(t) W(f) = U(f) V(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) IfV(f) consists entirely of Dirac impulses thenu(f) V(f) iust replaces each impulse with a complete copy ofu(f) scaled by the area of the impulse and shifted so thathz lies on the impulse. Then add the overlapping complex spectra. u(t) U(f) <U(f) (rad/pi a= Time (s) Frequency (Hz) Frequence (Hz) v(t) V(f) w(t) W(f) <W(f) (rad/pi Time (s).4 F= Frequency (Hz) a=2, F=2-5 5 Time (s) Frequency (Hz) Frequence (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 3 /
27 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
28 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
29 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Convolution in the time domain is equivalent to multiplication in the frequency domain and vice versa. E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
30 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Convolution in the time domain is equivalent to multiplication in the frequency domain and vice versa. Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) W(f) = U(f) V(f) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
31 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Convolution in the time domain is equivalent to multiplication in the frequency domain and vice versa. Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) W(f) = U(f) V(f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) X(f) = u( f) [duality] E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
32 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Convolution in the time domain is equivalent to multiplication in the frequency domain and vice versa. Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) W(f) = U(f) V(f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) X(f) = u( f) y(t) = V(t) Y(f) = v( f) [duality] E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
33 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Convolution in the time domain is equivalent to multiplication in the frequency domain and vice versa. Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) W(f) = U(f) V(f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) X(f) = u( f) y(t) = V(t) Y(f) = v( f) z(t) = W(t) Z(f) = w( f) [duality] E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
34 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Convolution in the time domain is equivalent to multiplication in the frequency domain and vice versa. Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) W(f) = U(f) V(f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) X(f) = u( f) y(t) = V(t) Y(f) = v( f) z(t) = W(t) Z(f) = w( f) [duality] Now the convolution property becomes: w( f) = u( f)v( f) W(t) = U(t) V(t) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
35 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Convolution in the time domain is equivalent to multiplication in the frequency domain and vice versa. Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) W(f) = U(f) V(f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) X(f) = u( f) y(t) = V(t) Y(f) = v( f) z(t) = W(t) Z(f) = w( f) [duality] Now the convolution property becomes: w( f) = u( f)v( f) W(t) = U(t) V(t) [subt ±f] E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
36 Convolution Convolution Parseval s Convolution : w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Convolution in the time domain is equivalent to multiplication in the frequency domain and vice versa. Proof of second line: Given u(t), v(t) and w(t) satisfying w(t) = u(t)v(t) W(f) = U(f) V(f) define dual waveforms x(t), y(t) and z(t) as follows: x(t) = U(t) X(f) = u( f) y(t) = V(t) Y(f) = v( f) z(t) = W(t) Z(f) = w( f) [duality] Now the convolution property becomes: w( f) = u( f)v( f) W(t) = U(t) V(t) [subt ±f] Z(f) = X(f)Y(f) z(t) = x(t) y(t) [duality] E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 4 /
37 Convolution Example Convolution Parseval s u(t) = t t < otherwise E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
38 Convolution Example Convolution Parseval s u(t) = t t < otherwise u(t).5 Time t (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
39 Convolution Example Convolution Parseval s u(t) = v(t) = t t < otherwise e t t t < u(t) Time t (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
40 Convolution Example Convolution Parseval s u(t) = v(t) = t t < otherwise e t t t < v(t) u(t) Time t (s) Time t (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
41 Convolution Example Convolution Parseval s u(t) = v(t) = t t < otherwise e t t t < v(t) u(t) Time t (s) Time t (s) w(t) = u(t) v(t) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
42 Convolution Example Convolution Parseval s u(t) = v(t) = t t < otherwise e t t t < v(t) u(t) Time t (s) Time t (s) w(t) = u(t) v(t) = u(τ)v(t τ)dτ E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
43 Convolution Example Convolution Parseval s u(t) = v(t) = t t < otherwise e t t t < v(t) u(t) Time t (s) Time t (s) w(t) = u(t) v(t) = u(τ)v(t τ)dτ = min(t,) ( τ)e τ t dτ E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
44 Convolution Example Convolution Parseval s u(t) = v(t) = t t < otherwise e t t t < v(t) u(t) Time t (s) Time t (s) w(t) = u(t) v(t) = u(τ)v(t τ)dτ = min(t,) ( τ)e τ t dτ = [(2 τ)e τ t ] min(t,) τ= E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
45 Convolution Example Convolution Parseval s u(t) = v(t) = t t < otherwise e t t t < v(t) u(t) Time t (s) Time t (s) w(t) = u(t) v(t) = u(τ)v(t τ)dτ = min(t,) ( τ)e τ t dτ = [(2 τ)e τ t ] min(t,) τ= t < = 2 t 2e t t < (e 2)e t t E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
46 Convolution Example Convolution Parseval s t t < u(t) = otherwise e t t v(t) = t < w(t) = u(t) v(t) = u(τ)v(t τ)dτ = min(t,) ( τ)e τ t dτ = [(2 τ)e τ t ] min(t,) τ= t < = 2 t 2e t t < (e 2)e t t u(t) v(t) w(t) Time t (s) Time t (s) Time t (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
47 Convolution Example Convolution Parseval s t t < u(t) = otherwise e t t v(t) = t < w(t) = u(t) v(t) = u(τ)v(t τ)dτ = min(t,) ( τ)e τ t dτ = [(2 τ)e τ t ] min(t,) τ= t < = 2 t 2e t t < (e 2)e t t u(t) v(t) w(t) = Time t (s) Time t (s) Time t (s) u(τ) v(.7-τ) t= Time τ (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
48 Convolution Example Convolution Parseval s t t < u(t) = otherwise e t t v(t) = t < w(t) = u(t) v(t) = u(τ)v(t τ)dτ = min(t,) ( τ)e τ t dτ = [(2 τ)e τ t ] min(t,) τ= t < = 2 t 2e t t < (e 2)e t t u(t) v(t) w(t) =.37.5 = Time t (s) Time t (s) Time t (s) u(τ) v(.7-τ) t= Time τ (s) u(τ) v(.5-τ) t= Time τ (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
49 Convolution Example Convolution Parseval s t t < u(t) = otherwise e t t v(t) = t < w(t) = u(t) v(t) = u(τ)v(t τ)dτ = min(t,) ( τ)e τ t dτ = [(2 τ)e τ t ] min(t,) τ= t < = 2 t 2e t t < (e 2)e t t u(t) v(t) w(t) =.37.5 =.6.5 = Time t (s) Time t (s) Time t (s) u(τ) v(.7-τ) t= Time τ (s) u(τ) v(.5-τ) t= Time τ (s) u(τ) v(2.5-τ) t= Time τ (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
50 Convolution Example Convolution Parseval s t t < u(t) = otherwise e t t v(t) = t < w(t) = u(t) v(t) = u(τ)v(t τ)dτ = min(t,) ( τ)e τ t dτ = [(2 τ)e τ t ] min(t,) τ= t < = 2 t 2e t t < (e 2)e t t u(t) v(t) w(t) =.37.5 =.6.5 = Time t (s) Time t (s) Time t (s) u(τ) v(.7-τ) t= Time τ (s) u(τ) v(.5-τ) t= Time τ (s) u(τ) v(2.5-τ) t= Time τ (s) Note howv(t τ) is time-reversed (because of the τ ) and time-shifted to put the time origin atτ = t. E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 5 /
51 Convolution Properties Convloution: w(t) = u(t) v(t) u(τ)v(t τ)dτ Convolution Parseval s E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 6 /
52 Convolution Properties Convolution Parseval s Convloution: w(t) = u(t) v(t) u(τ)v(t τ)dτ Convolution behaves algebraically like multiplication: ) Commutative: u(t) v(t) = v(t) u(t) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 6 /
53 Convolution Properties Convolution Parseval s Convloution: w(t) = u(t) v(t) u(τ)v(t τ)dτ Convolution behaves algebraically like multiplication: ) Commutative: u(t) v(t) = v(t) u(t) 2) Associative: u(t) v(t) w(t) = (u(t) v(t)) w(t) = u(t) (v(t) w(t)) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 6 /
54 Convolution Properties Convolution Parseval s Convloution: w(t) = u(t) v(t) u(τ)v(t τ)dτ Convolution behaves algebraically like multiplication: ) Commutative: u(t) v(t) = v(t) u(t) 2) Associative: u(t) v(t) w(t) = (u(t) v(t)) w(t) = u(t) (v(t) w(t)) 3) Distributive over addition: w(t) (u(t)+v(t)) = w(t) u(t)+w(t) v(t) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 6 /
55 Convolution Properties Convolution Parseval s Convloution: w(t) = u(t) v(t) u(τ)v(t τ)dτ Convolution behaves algebraically like multiplication: ) Commutative: u(t) v(t) = v(t) u(t) 2) Associative: u(t) v(t) w(t) = (u(t) v(t)) w(t) = u(t) (v(t) w(t)) 3) Distributive over addition: w(t) (u(t)+v(t)) = w(t) u(t)+w(t) v(t) 4) Identity Element or : u(t) δ(t) = δ(t) u(t) = u(t) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 6 /
56 Convolution Properties Convolution Parseval s Convloution: w(t) = u(t) v(t) u(τ)v(t τ)dτ Convolution behaves algebraically like multiplication: ) Commutative: u(t) v(t) = v(t) u(t) 2) Associative: u(t) v(t) w(t) = (u(t) v(t)) w(t) = u(t) (v(t) w(t)) 3) Distributive over addition: w(t) (u(t)+v(t)) = w(t) u(t)+w(t) v(t) 4) Identity Element or : u(t) δ(t) = δ(t) u(t) = u(t) 5) Bilinear: (au(t)) (bv(t)) = ab(u(t) v(t)) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 6 /
57 Convolution Properties Convolution Parseval s Convloution: w(t) = u(t) v(t) u(τ)v(t τ)dτ Convolution behaves algebraically like multiplication: ) Commutative: u(t) v(t) = v(t) u(t) 2) Associative: u(t) v(t) w(t) = (u(t) v(t)) w(t) = u(t) (v(t) w(t)) 3) Distributive over addition: w(t) (u(t)+v(t)) = w(t) u(t)+w(t) v(t) 4) Identity Element or : u(t) δ(t) = δ(t) u(t) = u(t) 5) Bilinear: (au(t)) (bv(t)) = ab(u(t) v(t)) Proof: In the frequency domain, convolution is multiplication. E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 6 /
58 Convolution Properties Convolution Parseval s Convloution: w(t) = u(t) v(t) u(τ)v(t τ)dτ Convolution behaves algebraically like multiplication: ) Commutative: u(t) v(t) = v(t) u(t) 2) Associative: u(t) v(t) w(t) = (u(t) v(t)) w(t) = u(t) (v(t) w(t)) 3) Distributive over addition: w(t) (u(t)+v(t)) = w(t) u(t)+w(t) v(t) 4) Identity Element or : u(t) δ(t) = δ(t) u(t) = u(t) 5) Bilinear: (au(t)) (bv(t)) = ab(u(t) v(t)) Proof: In the frequency domain, convolution is multiplication. Also, ifu(t) v(t) = w(t), then 6) Time Shifting: u(t+a) v(t+b) = w(t+a+b) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 6 /
59 Convolution Properties Convolution Parseval s Convloution: w(t) = u(t) v(t) u(τ)v(t τ)dτ Convolution behaves algebraically like multiplication: ) Commutative: u(t) v(t) = v(t) u(t) 2) Associative: u(t) v(t) w(t) = (u(t) v(t)) w(t) = u(t) (v(t) w(t)) 3) Distributive over addition: w(t) (u(t)+v(t)) = w(t) u(t)+w(t) v(t) 4) Identity Element or : u(t) δ(t) = δ(t) u(t) = u(t) 5) Bilinear: (au(t)) (bv(t)) = ab(u(t) v(t)) Proof: In the frequency domain, convolution is multiplication. Also, ifu(t) v(t) = w(t), then 6) Time Shifting: u(t+a) v(t+b) = w(t+a+b) 7) Time Scaling: u(at) v(at) = a w(at) How to recognise a convolution integral: the arguments ofu( ) andv( ) sum to a constant. E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 6 /
60 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
61 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
62 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
63 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
64 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
65 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt= δ(g f) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
66 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt= δ(g f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
67 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt= δ(g f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df Proof: Letu(t) = + f= U(f)ei2πft df and v(t) = + g= V(g)ei2πgt dg E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
68 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt= δ(g f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df Proof: Letu(t) = + f= U(f)ei2πft df and v(t) = + g= V(g)ei2πgt dg Now multiplyu (t) = u(t) andv(t) together and integrate over time: E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
69 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt= δ(g f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df Proof: Letu(t) = + f= U(f)ei2πft df and v(t) = + g= V(g)ei2πgt dg Now multiplyu (t) = u(t) andv(t) together and integrate over time: t= u (t)v(t)dt E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
70 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt= δ(g f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df Proof: Letu(t) = + f= U(f)ei2πft df and v(t) = + g= V(g)ei2πgt dg [Note use of different dummy variables] Now multiplyu (t) = u(t) andv(t) together and integrate over time: t= u (t)v(t)dt = t= + f= U (f)e i2πft df + g= V(g)ei2πgt dgdt E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
71 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt= δ(g f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df Proof: Letu(t) = + f= U(f)ei2πft df and v(t) = + g= V(g)ei2πgt dg [Note use of different dummy variables] Now multiplyu (t) = u(t) andv(t) together and integrate over time: t= u (t)v(t)dt = t= + f= U (f)e i2πft df + g= V(g)ei2πgt dgdt = + f= U (f) + g= V(g) t= ei2π(g f)t dtdgdf E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
72 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt= δ(g f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df Proof: Letu(t) = + f= U(f)ei2πft df and v(t) = + g= V(g)ei2πgt dg [Note use of different dummy variables] Now multiplyu (t) = u(t) andv(t) together and integrate over time: t= u (t)v(t)dt = t= + f= U (f)e i2πft df + g= V(g)ei2πgt dgdt = + f= U (f) + g= V(g) t= ei2π(g f)t dtdgdf = + f= U (f) + V(g)δ(g f)dgdf g= [lemma] E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
73 Parseval s Convolution Parseval s Lemma: X(f) = δ(f g) x(t) = δ(f g)e i2πft df = e i2πgt X(f) = e i2πgt e i2πft dt= e i2π(g f)t dt= δ(g f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df Proof: Letu(t) = + f= U(f)ei2πft df and v(t) = + g= V(g)ei2πgt dg [Note use of different dummy variables] Now multiplyu (t) = u(t) andv(t) together and integrate over time: t= u (t)v(t)dt = t= + f= U (f)e i2πft df + g= V(g)ei2πgt dgdt = + f= U (f) + g= V(g) t= ei2π(g f)t dtdgdf = + f= U (f) + V(g)δ(g f)dgdf g= [lemma] = + f= U (f)v(f)df E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 7 /
74 Energy Conservation Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df Convolution Parseval s E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
75 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
76 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
77 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df Energy Conservation: The energy inu(t) equals the energy inu(f). E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
78 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df Energy Conservation: The energy inu(t) equals the energy inu(f). Example: u(t) = e at t t < u(t) a= Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
79 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df Energy Conservation: The energy inu(t) equals the energy inu(f). Example: u(t) = e at t t < E u = u(t) 2 dt = [ ] e 2at 2a u(t) a= Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
80 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df Energy Conservation: The energy inu(t) equals the energy inu(f). Example: u(t) = e at t t < E u = u(t) 2 dt = [ ] e 2at 2a = 2a u(t) a= Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
81 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df Energy Conservation: The energy inu(t) equals the energy inu(f). Example: u(t) = e at t t < E u = u(t) 2 dt = [ ] e 2at 2a = 2a U(f) = a+i2πf [from before] u(t) U(f) a= Time (s) Frequency (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
82 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df Energy Conservation: The energy inu(t) equals the energy inu(f). Example: u(t) = e at t t < E u = u(t) 2 dt = [ ] e 2at 2a = 2a U(f) = a+i2πf U(f) 2 df = [from before] u(t) a=2.5 df -5 5 a 2 +4π 2 f 2 Time (s) U(f) Frequency (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
83 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df Energy Conservation: The energy inu(t) equals the energy inu(f). Example: u(t) = e at t t < E u = u(t) 2 dt = [ ] e 2at 2a = 2a U(f) = a+i2πf U(f) 2 df = [ tan = ( 2πf a ) 2πa df a 2 +4π 2 f ] 2 [from before] u(t) U(f) a= Time (s) Frequency (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
84 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df Energy Conservation: The energy inu(t) equals the energy inu(f). Example: u(t) = e at t t < E u = u(t) 2 dt = [ ] e 2at 2a = 2a U(f) = a+i2πf U(f) 2 df = [ tan = ( 2πf a ) 2πa df a 2 +4π 2 f ] 2 = π 2πa [from before] u(t) U(f) a= Time (s) Frequency (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
85 Energy Conservation Convolution Parseval s Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df For the special case v(t) = u(t), Parseval s theorem becomes: t= u (t)u(t)dt = + f= U (f)u(f)df E u = t= u(t) 2 dt = + f= U(f) 2 df Energy Conservation: The energy inu(t) equals the energy inu(f). Example: u(t) = e at t t < E u = u(t) 2 dt = [ ] e 2at 2a = 2a U(f) = a+i2πf U(f) 2 df = [ tan = ( 2πf a ) 2πa df a 2 +4π 2 f ] 2 [from before] = π 2πa = 2a u(t) U(f) a= Time (s) Frequency (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 8 /
86 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
87 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < w(t) a=2, F=2-5 5 Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
88 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < W(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) w(t) a=2, F=2-5 5 Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
89 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < W(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) = a+i2πf a 2 +i4πaf 4π 2 (f 2 F 2 ) w(t) a=2, F=2-5 5 Time (s) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
90 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < W(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) = a+i2πf a 2 +i4πaf 4π 2 (f 2 F 2 ) w(t) W(f) <W(f) (rad/pi a=2, F=2-5 5 Time (s) Frequency (Hz) Frequence (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
91 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < W(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) = a+i2πf a 2 +i4πaf 4π 2 (f 2 F 2 ) w(t) W(f) <W(f) (rad/pi a=2, F=2-5 5 Time (s) Frequency (Hz) Frequence (Hz) W(f) 2 = a 2 +4π 2 f 2 (a 2 4π 2 (f 2 F 2 )) 2 +6π 2 a 2 f 2 E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
92 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < W(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) = W(f) 2 = a+i2πf a 2 +i4πaf 4π 2 (f 2 F 2 ) a 2 +4π 2 f 2 (a 2 4π 2 (f 2 F 2 )) 2 +6π 2 a 2 f 2 w(t) W(f) <W(f) (rad/pi W(f) a=2, F=2-5 5 Time (s) Frequency (Hz) Frequence (Hz) Frequency (Hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
93 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < W(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) = W(f) 2 = a+i2πf a 2 +i4πaf 4π 2 (f 2 F 2 ) a 2 +4π 2 f 2 (a 2 4π 2 (f 2 F 2 )) 2 +6π 2 a 2 f 2 w(t) W(f) <W(f) (rad/pi W(f) a=2, F=2-5 5 Time (s) Frequency (Hz) Frequence (Hz) Frequency (Hz) The units of W(f) 2 are energy per Hz so that its integral, E w = W(f) 2 df, has units of energy. E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
94 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < W(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) = W(f) 2 = a+i2πf a 2 +i4πaf 4π 2 (f 2 F 2 ) a 2 +4π 2 f 2 (a 2 4π 2 (f 2 F 2 )) 2 +6π 2 a 2 f 2 w(t) W(f) <W(f) (rad/pi W(f) a=2, F=2-5 5 Time (s) Frequency (Hz) Frequence (Hz) Frequency (Hz) Energy Spectrum The units of W(f) 2 are energy per Hz so that its integral, E w = W(f) 2 df, has units of energy. The quantity W(f) 2 is called the energy spectral density ofw(t) at frequency f and its graph is the energy spectrum ofw(t). E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
95 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < W(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) = W(f) 2 = a+i2πf a 2 +i4πaf 4π 2 (f 2 F 2 ) a 2 +4π 2 f 2 (a 2 4π 2 (f 2 F 2 )) 2 +6π 2 a 2 f 2 w(t) W(f) <W(f) (rad/pi W(f) a=2, F=2-5 5 Time (s) Frequency (Hz) Frequence (Hz) Frequency (Hz) Energy Spectrum The units of W(f) 2 are energy per Hz so that its integral, E w = W(f) 2 df, has units of energy. The quantity W(f) 2 is called the energy spectral density ofw(t) at frequency f and its graph is the energy spectrum ofw(t). It shows how the energy of w(t) is distributed over frequencies. E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
96 Energy Spectrum Convolution Parseval s Example from before: w(t) = e at cos2πft t t < W(f) =.5 a+i2π(f+f) +.5 a+i2π(f F) = W(f) 2 = a+i2πf a 2 +i4πaf 4π 2 (f 2 F 2 ) a 2 +4π 2 f 2 (a 2 4π 2 (f 2 F 2 )) 2 +6π 2 a 2 f 2 w(t) W(f) <W(f) (rad/pi W(f) a=2, F=2-5 5 Time (s) Frequency (Hz) Frequence (Hz) Frequency (Hz) Energy Spectrum The units of W(f) 2 are energy per Hz so that its integral, E w = W(f) 2 df, has units of energy. The quantity W(f) 2 is called the energy spectral density ofw(t) at frequency f and its graph is the energy spectrum ofw(t). It shows how the energy of w(t) is distributed over frequencies. If you divide W(f) 2 by the total energy,e w, the result is non-negative and integrates to unity like a probability distribution. E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 9 /
97 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
98 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
99 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot Acts like multiplication + time scaling/shifting formulae E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
100 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot Acts like multiplication + time scaling/shifting formulae Convolution : multiplication convolution E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
101 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot Acts like multiplication + time scaling/shifting formulae Convolution : multiplication convolution w(t) = u(t)v(t) W(f) = U(f) V(f) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
102 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot Acts like multiplication + time scaling/shifting formulae Convolution : multiplication convolution w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
103 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot Acts like multiplication + time scaling/shifting formulae Convolution : multiplication convolution w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
104 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot Acts like multiplication + time scaling/shifting formulae Convolution : multiplication convolution w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df : E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
105 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot Acts like multiplication + time scaling/shifting formulae Convolution : multiplication convolution w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df : Energy spectral density: U(f) 2 (energy/hz) E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
106 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot Acts like multiplication + time scaling/shifting formulae Convolution : multiplication convolution w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df : Energy spectral density: U(f) 2 (energy/hz) Parseval: E u = u(t) 2 dt = U(f) 2 df E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
107 Summary Convolution Parseval s Convolution: u(t) v(t) u(τ)v(t τ)dτ Arguments ofu( ) andv( ) sum tot Acts like multiplication + time scaling/shifting formulae Convolution : multiplication convolution w(t) = u(t)v(t) W(f) = U(f) V(f) w(t) = u(t) v(t) W(f) = U(f)V(f) Parseval s : t= u (t)v(t)dt = + f= U (f)v(f)df : Energy spectral density: U(f) 2 (energy/hz) Parseval: E u = u(t) 2 dt = U(f) 2 df For further details see RHB Chapter 3. E. Fourier Series and Transforms ( ) Fourier Transform - Parseval and Convolution: 7 /
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