 # APPLICATIONS OF FOURIER REPRESENTATIONS TO

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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:

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?

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

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

E Example 4.1, p343:

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

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:

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?

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

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

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

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:

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).

Thus,

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.

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: