1

Signals and Systems Lecture (6)

Fourier Transforms

Date April 4, 2008

Today’s Topics 1. The Fourier transform 2. Examples Take Away

The extension of Fourier series to the Fourier transform provides a means to analyze aperiodic signals.

Required Reading

O&W-4.1.1, 4.1.2, 4.1.3 (Example 4.1), 4.2

2

The Fourier series approach to the analysis of signals and systems is very useful in terms of gaining insight about the nature of signals as sums of sinusoidal functions. In particular, we learned that a very large class of signals can be represented, to any desired level of accuracy, by sums of sinusoidal functions. As well, we developed the concept of frequency response and demonstrated its utility in relating the output to the input of a LTI system. However, the periodicity requirement that we had to impose, in order to obtain Fourier series representations of signals, is restrictive. In particular it restricts our ability to explore the larger class of aperiodic signals, which are very important for aerospace systems. It turns out that in the process of removing the requirement for periodicity we will develop an operator algebra that will be invaluable for understanding and analyzing many LTI systems. The first step in this process is the development of the Fourier transform, which is essential for understanding communication systems. Subsequently we will develop its successor, the Laplace transform, which is a enormously useful for analyzing and designing control systems.

3

Fourier Transform Suppose we have some signal

!

x(t) that is of finite duration, meaning that it is zero for all time before some initial time and is zero for all time after some final time. Furthermore, the duration of the signal is less than a time T. An example of such a signal is illustrated in the following figure.

We have chosen to define our time origin at the midpoint between the beginning and the end if the signal duration. Hence the signal must begin after the time –T/2 and must end before the time +T/2. We now replicate the signal at intervals of +T to create the signal

!

˜ x (t) . Thus, by construction, the signal

!

˜ x (t) is identical to

!

x(t) in the interval +T/2 and is periodic with period T, as shown in he following diagram.

4

Assuming that all of the Dirichlet conditions are satisfied, there is a Fourier series representation of

!

˜ x (t) as follows-

Also, from our construction of

!

˜ x (t) we know that

and if we define a complex function of the continuous variable

!

" as

then the Fourier coefficients for

!

˜ x (t) can we written as

5

In anticipation of eventually taking a limit, we define the following incremental change in frequency

and for any value of the index k we define frequency as a multiple of the increment

!

"#

so the equation for the Fourier coefficients becomes

and substituting back into the equation for the Fourier series for

!

˜ x (t) obtains

or, equivalently since

!

T = (2" /#0) = (2" /$#)

6

Now

!

k"# =# , and if we let

!

T"# then

!

"# = 2$ /T% 0 , and in the limit the equation for

!

˜ x (t) becomes an integral

Furthermore, as

!

T"# then

!

˜ x (t)" x(t) , so we have the Fourier transform pair

The function

!

X( j") is the Fourier transform of the signal x(t) and conversely x(t) is the inverse Fourier transform of

!

X( j"). Since we allowed the period T to go to infinity the signal x(t) can be of arbitrary duration and therefore aperiodic. Also, if x(t) is a system impulse response then its Fourier transform is the system frequency response.

7

In designating a Fourier transform pair, which is to say the signal x(t) and its Fourier transform

!

X( j"), we will often use the following notation

Conditions to Ensure Convergence of Fourier Transforms As was the case for Fourier series, the Dirichlet conditions ensure that a signal x(t) has a Fourier transform

x(t) be absolutely integrable x(t) have a finite number of maxima and minima in any finite interval x(t) have a finite number of discontinuities in any finite interval and discontinuities must be finite

8

Example-Fourier Transform of an Exponential The function x(t) has an exponential so impulse response

Note, in particular, that this function is greater than zero for all positive values of time and approaches zero asymptotically. Hence it is aperiodic, because there is no finite period, and it does not have a Fourier series representation. However, its Fourier transform is

Thus, once again we have the first order system frequency response, with magnitude and phase-

9

which are the familiar equations for the frequency response of a first order system with time constant

!

" =1/a, as depicted in these familiar diagrams

10

Fourier Transform of a Pulse of Unit Area in the Time Domain

11

12

Fourier Transform of an Impulse in the Time Domain

13

Inverse Fourier Transform of a Pulse of Unit Area in the Frequency Domain

14

Inverse Fourier Transform of an Impulse in the Frequency Domain

15

Fourier Transform of a Unit Step The unit step is not absolutely integrable so its Fourier transform cannot be obtained by direct integration. We will obtain its Fourier transform as a limit of exponential functions.

16

17

Inverse Fourier Transform of Real Impulses at

!

±

!

"0 in

the Frequency Domain

18

Inverse Fourier Transform of Imaginary Impulses in the Frequency Domain at

!

±

!

"0

19

Fourier Transforms of Periodic Functions