7: Fourier Transforms: Convolution and Parseval’s Theorem

1 7: Fourier Transforms: Convolution and Parseval's Theorem Multiplication of Signals Multiplication Example Convolution Theorem Convolution Example Convolution properties Parseval's Theorem energy Conservation energy Spectrum Summary 7: Fourier Transforms: Convolution and Parseval's Theorem Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 1 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem Multiplication of Signals Multiplication Example Convolution Theorem Convolution Example Convolution properties Parseval's Theorem energy Conservation energy Spectrum Summary Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g=.

2 V (g)e dg Multiplication Example Convolution Theorem Convolution Example Convolution properties Parseval's Theorem energy Conservation energy Spectrum Summary Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem energy Conservation energy Spectrum Summary Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem [Note use of different dummy variables].

3 Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem R + i2 ht R + . energy Conservation energy Spectrum = h= U (h)e dh g= V (g)ei2 gt dg Summary Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem [Note use of different dummy variables]. Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem R + i2 ht R + . energy Conservation energy Spectrum = h= U (h)e dh g= V (g)ei2 gt dg Summary R + R + . = h= U (h) g= V (g)ei2 (h+g)t dg dh [merge e( ) ]. Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ?

4 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem [Note use of different dummy variables]. Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem R + i2 ht R + . energy Conservation energy Spectrum = h= U (h)e dh g= V (g)ei2 gt dg Summary R + R + . = h= U (h) g= V (g)ei2 (h+g)t dg dh [merge e( ) ]. Now we make a change of variable in the second integral: g = f h R + R + . = h= U (h) f = V (f h)ei2 f t df dh Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem [Note use of different dummy variables].

5 Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem R + i2 ht R + . energy Conservation energy Spectrum = h= U (h)e dh g= V (g)ei2 gt dg Summary R + R + . = h= U (h) g= V (g)ei2 (h+g)t dg dh [merge e( ) ]. Now we make a change of variable in the second integral: g = f h R + R + . = h= U (h) f = V (f h)ei2 f t df dh R R + i2 f t R. = f = h= U (h)V (f h)e dh df [swap ]. Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem [Note use of different dummy variables]. Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem R + i2 ht R + . energy Conservation energy Spectrum = h= U (h)e dh g= V (g)ei2 gt dg Summary R + R +.

6 = h= U (h) g= V (g)ei2 (h+g)t dg dh [merge e( ) ]. Now we make a change of variable in the second integral: g = f h R + R + . = h= U (h) f = V (f h)ei2 f t df dh R R + i2 f t R. = f = h= U (h)V (f h)e dh df [swap ]. R + . = f = W (f )ei2 f t df Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem [Note use of different dummy variables]. Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem R + i2 ht R + . energy Conservation energy Spectrum = h= U (h)e dh g= V (g)ei2 gt dg Summary R + R + . = h= U (h) g= V (g)ei2 (h+g)t dg dh [merge e( ) ]. Now we make a change of variable in the second integral: g = f h R + R + . = h= U (h) f = V (f h)ei2 f t df dh R R + i2 f t R.

7 = f = h= U (h)V (f h)e dh df [swap ]. R + . = f = W (f )ei2 f t df R + . where W (f ) = h= U (h)V (f h)dh Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem [Note use of different dummy variables]. Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem R + i2 ht R + . energy Conservation energy Spectrum = h= U (h)e dh g= V (g)ei2 gt dg Summary R + R + . = h= U (h) g= V (g)ei2 (h+g)t dg dh [merge e( ) ]. Now we make a change of variable in the second integral: g = f h R + R + . = h= U (h) f = V (f h)ei2 f t df dh R R + i2 f t R. = f = h= U (h)V (f h)e dh df [swap ]. R + . = f = W (f )ei2 f t df R + . where W (f ) = h= U (h)V (f h)dh , U (f ) V (f ).

8 Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10. Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem [Note use of different dummy variables]. Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem R + i2 ht R + . energy Conservation energy Spectrum = h= U (h)e dh g= V (g)ei2 gt dg Summary R + R + . = h= U (h) g= V (g)ei2 (h+g)t dg dh [merge e( ) ]. Now we make a change of variable in the second integral: g = f h R + R + . = h= U (h) f = V (f h)ei2 f t df dh R R + i2 f t R. = f = h= U (h)V (f h)e dh df [swap ]. R + . = f = W (f )ei2 f t df R + . where W (f ) = h= U (h)V (f h)dh , U (f ) V (f ). This is the Convolution of the two spectra U (f ) and V (f ). Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10.

9 Multiplication of Signals Question: What is the Fourier transform of w(t) = u(t)v(t) ? 7: Fourier Transforms: Convolution and Parseval's Theorem R + R + . i2 ht i2 gt Multiplication of Signals Let u(t) = h= U (h)e dh and v(t) = g= . V (g)e dg Multiplication Example Convolution Theorem [Note use of different dummy variables]. Convolution Example Convolution properties w(t) = u(t)v(t). Parseval's Theorem R + i2 ht R + . energy Conservation energy Spectrum = h= U (h)e dh g= V (g)ei2 gt dg Summary R + R + . = h= U (h) g= V (g)ei2 (h+g)t dg dh [merge e( ) ]. Now we make a change of variable in the second integral: g = f h R + R + . = h= U (h) f = V (f h)ei2 f t df dh R R + i2 f t R. = f = h= U (h)V (f h)e dh df [swap ]. R + . = f = W (f )ei2 f t df R + . where W (f ) = h= U (h)V (f h)dh , U (f ) V (f ). This is the Convolution of the two spectra U (f ) and V (f ). w(t) = u(t)v(t) W (f ) = U (f ) V (f ). Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 2 / 10.

10 Multiplication Example 7: Fourier Transforms: ( 1. at Convolution and Parseval's e t 0 a=2. u(t). Theorem u(t) = 0. Multiplication of Signals Multiplication Example 0 t<0 -5 0. Time (s). 5. Convolution Theorem Convolution Example Convolution properties Parseval's Theorem energy Conservation energy Spectrum Summary Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 3 / 10. Multiplication Example 7: Fourier Transforms: ( 1. at Convolution and Parseval's e t 0 a=2. u(t). Theorem u(t) = 0. Multiplication of Signals Multiplication Example 0 t<0 -5 0. Time (s). 5. Convolution Theorem 1. Convolution Example U (f ) = a+i2 f [from before]. Convolution properties Parseval's Theorem energy Conservation energy Spectrum Summary Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution : 7 3 / 10. Multiplication Example 7: Fourier Transforms: ( 1. at Convolution and Parseval's e t 0 a=2. u(t). Theorem u(t) = 0. Multiplication of Signals Multiplication Example 0 t<0 -5 0.)))