1. APPLICATIONS OF FOURIER REPRESENTATIONS TO
CHAPTER 4
APPLICATIONS OF FOURIER REPRESENTATIONS TO
MIXED SIGNAL CLASSES
What about the Fourier representation of a mixture of
a) periodic and non-periodic signals
b) CT and DT signals?
Examples:

2. We will go through:
a) FT of periodic signals, which we have used FS:
We can take FT of x(t).
b) Convolution and multiplication with mixture of periodic and
non-periodic signals.
c) Fourier transform of discrete-time signals.
FT of periodic signals
Chapter 3: for CT periodic signals, FS representations.
What happens if we take FT of periodic signals?

3. FS representation of periodic signal x(t):
Take FT of equation (*) 
Note:
a) FT of a periodic signal is a series of impulses spaced by the
fundamental frequency w0.
b) The k-th impulse has strength 2pX[k].
c) FT of x(t)=cos(w0t) can be obtained by replacing

4. FS and FT representation of a periodic continuous-time signal.

5. E
Example 4.1, p343:

6. E
Example 4.2, p344:
p(t) is periodic with fundamental period T, fundamental frequency w0. FS coefficients:

7. Relating DTFT to DTFS N-periodic signal x[n] has DTFS expression
Extending to any interval:
This, DTFT of x[n] given in (*) is expressed as:

8. E Problem 4.3(c), p347: Since X[k] is N periodic and NW0=2p, we have
Note:
a) DTFS  DTFT:
b) DTFT  DTFS:
Also, replace sum intervals from 0~N-1 for DTFS to - ~  for DTFT E
Problem 4.3(c), p347:
Fundamental period?

9. Use note a) last slide:
Question: if we take inverse DTFS of X[k], we get
Exercise: use Matlab to verify.

10. Convolution and multiplication with mixture of periodic
and non-periodic signals
For periodic inputs:
1) Convolution of periodic and non-periodic signals

11. E
Problem 4.4(a), p350: LTI system
has an impulse response

12. Because h(t) is an ideal bandpass filter with a bandwidth 2p centered at 4p, the Fourier transform of the output signal is thus
which has a time-domain expression given as:
For discrete-time signals:

13. 2) Multiplication of periodic and non-periodic signals
Carrying out the convolution yields:
DT case: E
Problem 4.7, p357(b):
Consider the LTI system and input signal spectrum X(ejW) depicted by the figure below. Determine an expression for Y(ejW), the DTFT of the output y[n] assuming that z[n]=2cos(pn/2).

14. Thus,

15. E
Example 4.6, p353: AM Radio
(a) Simplified AM radio transmitter & receiver. (b) Spectrum of message signal Analyze the system in the frequency domain.

16. After low-pass filtering:
Signals in the AM transmitter and receiver. (a) Transmitted signal r(t) and spectrum R(j). (b) Spectrum of q(t) in the receiver. (c) Spectrum of receiver output y(t).
In the receiver, r(t) is multiplied by the identical cosine used in the transmitter to obtain:
After low-pass filtering: