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CHAPTER 4 APPLICATIONS OF FOURIER REPRESENTATIONS TO MIXED SIGNAL CLASSES

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:.

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CHAPTER 4 APPLICATIONS OF FOURIER REPRESENTATIONS TO MIXED SIGNAL CLASSES

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  1. 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 impulsehas 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. Example 4.1, p343: E

  6. Example 4.2, p344: E 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. 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 Problem 4.3(c), p347: E 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. Problem 4.4(a), p350: LTI system has an impulse response E

  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. 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:

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