1. ENEE 322: Continuous-Time Fourier Transform (Chapter 4)
Richard J. La
Spring 2016

2. Motivation
Recall: Continuous-time Fourier series representation allows us to write a large class of periodic signals as a linear combination of complex exponentials with harmonic frequencies
Most of real signals are not periodic
Usefulness of the idea below will be severely limited if aperiodic signals could not be written in terms of complex exponentials
LTI with IR h(t)

3. Continuous-time Fourier Transform (1)
It turns out a large class of aperiodic signals can also be written as linear combination of complex exponentials
Suppose that a continuous-time signal x(t) is absolutely integrable, i.e.,
Fourier transform:
Inverse Fourier transform:

4. Continuous-time Fourier Transform (2) *
Interpretation of inverse Fourier transform
Let denote the set of all complex-valued continuous-time signals with finite energy
Inner product of two signals is given by
Fourier transform:
where

5. Continuous-time Fourier Transform (3) *
Substituting last expression in the inverse Fourier transform
In turns out is an orthogonal basis for
Fourier transform of a signal x(t) (with finite energy) tells us how the energy of the signal is spread in the frequency domain
In the same way the Fourier series representation tells us how the average power of a periodic signal is spread in the frequency domain

6. Continuous-time Fourier Transform (4)
Fourier transform pair:
There are many ways to represent a signal
Signal is the representation in time domain
Fourier transform is just another representation of the same signal in frequency domain
Allows us to go back and forth between time domain and frequency domain

7. Fourier Transform of periodic signals
Fourier transform of a periodic signal with period T
Example #1:

8. Examples (2)
Example #2: Example #3: Example #4:

9. Properties of continuous-time Fourier transform (1)
Linearity:
Time shifting:
Time reversal:
Complex conjugation:
Special case: x(t) is real –
What does it say about real and imaginary parts of the Fourier transform?

10. Properties of continuous-time Fourier transform (2)
Parseval’s relation:
Differentiation:
Integration:
Time & frequency scaling:

11. Properties of continuous-time Fourier transform (3)
Duality (*)
Oftentimes we know and
We can compute y(t) quite easily without performing inverse Fourier transform

12. Properties of continuous-time Fourier transform (4)
Example #1:
What is the signal y(t) that has Fourier transform given below?

13. Properties of continuous-time Fourier transform (5)
Example #2: Differentiation in frequency domain
Example #3: Shifting in frequency domain (often used in communication systems due to regulation)

14. Properties of continuous-time Fourier transform (6)
Example #5: Integration in frequency domain

15. Convolution & Multiplication Property (1)
Recall: Take a complex exponential and pass it through an LTI system with impulse response h(t)
where
Guess what would be the output when the input is given by
LTI with IR
h(t)
Fourier transform of h(t)
evaluated at !!!!

16. Convolution & Multiplication Property (2)
Earlier with weighted sum of complex exponentials
where
Now
LTI with IR
h(t)
LTI with IR
h(t)
????

17. Convolution & Multiplication Property (3)
Answer: Drum roll please ….
Look at the expression for the output carefully
LTI with IR h(t)
inverse Fourier transform

18. Convolution & Multiplication Property (4)
Second equality tells us ….
Fourier transform of the output is given by the product of the Fourier transform of the input signal and that of the impulse response
Gives us another method for computing the output (in time domain) given an input signal and the impulse response
Compute the Fourier transform of the input signal and the impulse response (called “frequency response”)
Multiply the Fourier transforms to obtain the Fourier transform of the output
Take the inverse Fourier transform to obtain the output in time domain

19. Convolution & Multiplication Property (5)
Convolution in time domain Multiplication in frequency domain
The Fourier transform of the impulse response called the “frequency response” of the system
Tells us how each component with different frequency is changed by the system in magnitude and phase
Caveat: Since the Fourier transform is not defined for all signals, the frequency response is not defined for all LTI systems.
We will assume that the frequency response of the LTI systems we deal with exists

20. Convolution & Multiplication Property (6)
Implication for cascade systems
The frequency response of a cascade of LTI systems is given by the product of the frequency responses of the LTI systems S1 S2

21. Convolution & Multiplication Property (5)
What if we have a multiplication in time domain?
Multiplication in time domain becomes convolution in frequency domain

22. Examples (1)
Example : Suppose that we have an LTI system with impulse response Find the output when the input x(t) is given by (a) (b)