Transcript Convolution, Fourier Series, and the Fourier Transform

Convolution

A mathematical operator which computes the pointwise overlap
between two functions.

Discrete domain:

Continuous domain:
Discrete domain
Basic steps
1.
2.
3.
4.
5.
Flip (reverse) one of the functions.
Shift it along the time axis by one sample.
Pointwise multiply the corresponding values of the two digital
functions.
Sum the products from step 3 to get one point of the digital
convolution.
Repeat steps 1-4 to obtain the digital convolution at all times that the
functions overlap.
Continuous domain example
Continuous domain example
LTI (Linear Time-Invariant) Systems

Convolution can describe the effect of an LTI system on a signal

Assume we have an LTI system H, and its impulse response h[n]

Then if the input signal is x[n], the output signal is y[n] = x[n] * h[n]
x[n]
H
y[n] = x[n]*h[n]
Fourier Series

Most periodic functions can be expressed as a (infinite) linear
combination of sines and cosines
F(t) = a0 + a1cos (ωt) + b1sin(ωt) +
a2cos (2ωt) + b2sin(2ωt) + …
=


n 0
(an cos( nt ) bn sin( nt ))
F(t) is a periodic function with

2

T
Most Functions?

F(t) is a periodic function with

must satisfy certain other conditions:
–
–
–
finite number of discontinuities within T
finite average within T
finite number of minima and maxima

2

T
Calculate Coefficients
F (t )  n0 (an cos( nt ) bn sin( nt ))

T
a0 
 f (t )dt
0
2
ak 
T
T
T
 f (t ) cos(kt )dt
0
2
bk 
T
T
 f (t ) sin( kt )dt
0
Example

F(t) = square wave, with T=1.0s (

)
2

1.0
Series1
1
0
0
1
2
3
4
5
6
7
Example
F (t ) 
4

sin t
Series1
Series2
1
0
0
1
2
3
4
5
6
7
Example
4
4
4
F (t )  sin t 
sin 3t  sin 5t

3
5
Series1
Series3
1
0
0
1
2
3
4
5
6
7
Why Frequency Domain?

Allows efficient representation of a good approximation to the
original function

Makes filtering easy

And a whole host of other reasons….
But One Really Important One

Note that convolution in the time domain is equivalent to
multiplication in the frequency domain (and vice versa)!
Fourier Family
Fourier Transform

Definitions:

Can be difficult to compute => Often rely upon table of transforms
Delta function

Definition:

Often, the result of the Fourier Transform needs to be expressed in
terms of the delta function
Fourier Transform pairs





There is a duality in all transform pairs
