Analysis and Transmission of Signals in Communication Systems

1. 1 C H A P T E R 3 ANALYSIS AND TRANSMISSION OF SIGNALS

2. 2 Fundamental of Communication Systems ELCT332 Fall2011 Figure 3.1 Construction of a periodic signal by periodic extension of g(t) . Aperiodic Signal: Fourier Integral

3. 3 Fundamental of Communication Systems ELCT332 Fall2011 Figure 3.2 Change in the Fourier spectrum when the period T 0 in Fig. 3.1 is doubled.

4. 4 Fundamental of Communication Systems ELCT332 Fall2011 The Fourier series becomes the Fourier integral in the limit as T 0 .

5. 5 Fundamental of Communication Systems ELCT332 Fall2011 (a) e at u(t) and (b) its Fourier spectra. Fourier integral G(f): direct Fourier transform of g(t) g(t): inverse Fourier transform of G(f) Find the Fourier transform of Dirichlet Condition Linearity of the Fourier Transform (Superposition Theorem)

6. 6 Fundamental of Communication Systems ELCT332 Fall2011 Analogy for Fourier transform. Physical Appreciation of the Fourier Transform Fourier representation is a way of a signal in terms of everlasting sinusoids, or exponentials. The Fourier Spectrum of a signal indicates the relative amplitudes and phases of the sinusoids that are required to synthesize the signal.

7. 7 Fundamental of Communication Systems ELCT332 Fall2011 Time-limited pulse. G(f): Spectrum of g(t)

8. 8 Fundamental of Communication Systems ELCT332 Fall2011 Rectangular pulse. Unit Rectangular Function Transforms of some useful functions

9. 9 Fundamental of Communication Systems ELCT332 Fall2011 Triangular pulse. Unit Triangular Function

10. 10 Fundamental of Communication Systems ELCT332 Fall2011 Sinc pulse. Sinc Function

11. 11 Fundamental of Communication Systems ELCT332 Fall2011 (a) Rectangular pulse and (b) its Fourier spectrum. Example

12. 12 Fundamental of Communication Systems ELCT332 Fall2011 (a) Unit impulse and (b) its Fourier spectrum. Example II

13. 13 Fundamental of Communication Systems ELCT332 Fall2011 (a) Constant (dc) signal and (b) its Fourier spectrum. Example III

14. 14 Fundamental of Communication Systems ELCT332 Fall2011 (a) Cosine signal and (b) its Fourier spectrum. Find the inverse Fourier transform of

15. 15 Fundamental of Communication Systems ELCT332 Fall2011 Sign function.

16. 16 Fundamental of Communication Systems ELCT332 Fall2011 Near symmetry between direct and inverse Fourier transforms. Time-Frequency Duality Dual Property

17. 17 Fundamental of Communication Systems ELCT332 Fall2011 Duality property of the Fourier transform. Dual Property

18. 18 Fundamental of Communication Systems ELCT332 Fall2011 The scaling property of the Fourier transform. Time-Scaling Property Time compression of a signal results in spectral expansion, and time expansion of the signal results in its spectral compression.

19. 19 Fundamental of Communication Systems ELCT332 Fall2011 (a) e a | t | and (b) its Fourier spectrum. Example Prove that and if to find the Fourier transforms of and

20. 20 Fundamental of Communication Systems ELCT332 Fall2011 Physical explanation of the time-shifting property. Time-Shifting Property Delaying a signal by t 0 seconds does not change its amplitude spectrum. The phase spectrum is changed by -2 ft 0 . To achieve the same time delay, higher frequency sinusoids must undergo proportionately larger phase shifts. Question: Prove that

21. 21 Fundamental of Communication Systems ELCT332 Fall2011 Effect of time shifting on the Fourier spectrum of a signal. Example Find the Fourier transform of Linear phase spectrum

22. 22 Fundamental of Communication Systems ELCT332 Fall2011 Amplitude modulation of a signal causes spectral shifting. Frequency-Shifting Property Multiplication of a signal by a factor shifts the spectrum of that signal by f=f 0 Amplitude Modulation Carrier, Modulating signal, Modulated signal

23. 23 Fundamental of Communication Systems ELCT332 Fall2011 Example of spectral shifting by amplitude modulation. Example: Find the Fourier transform of the modulated signal g(t)cos2 f 0 t in which g(t) is a rectangular pulse Frequency division multiplexing (FDM)

24. 24 Fundamental of Communication Systems ELCT332 Fall2011 (a) Bandpass signal and (b) its spectrum. Bandpass Signals

25. 25 Fundamental of Communication Systems ELCT332 Fall2011 (a) Impulse train and (b) its spectrum. Example: Find the Fourier transform of a general periodic signal g(t) of period T 0

26. 26 Fundamental of Communication Systems ELCT332 Fall2011 Using the time differentiation property to find the Fourier transform of a piecewise-linear signal. Time Differentiation Time Integration Find the Fourier transform of the triangular pulse

27. 27 Fundamental of Communication Systems ELCT332 Fall2011 Properties of Fourier Transform Operations Operation g(t) G(f) Superposition g1(t)+g2(t) G1(f)+G2(f) Scalar multiplication kg(t) kG(f) Duality G(t) g(-f) Time scaling g(at) Time shifting g(t-t0) Frequency Shift G(f-f0) Time convolution g1(t)*g2(t) G1(f)G2(f) Frequency convolution g1(t)g2(t) G1(f)*G2(f) Time differentiation Time integration

28. 28 Fundamental of Communication Systems ELCT332 Fall2011 Signal transmission through a linear time-invariant system. H(f): Transfer function/frequency response Signal Transmission Through a Linear System

29. 29 Fundamental of Communication Systems ELCT332 Fall2011 Linear time invariant system frequency response for distortionless transmission. Distortionless transmission : a signal to pass without distortion delayed ouput retains the waveform

30. 30 Fundamental of Communication Systems ELCT332 Fall2011 (a) Simple RC filter. (b) Its frequency response and time delay. Determine the transfer function H(f), and td(f). What is the requirement on the bandwidth of g(t) if amplitude variation within 2% and time delay variation within 5% are tolerable?

31. 31 Fundamental of Communication Systems ELCT332 Fall2011 (a) Ideal low-pass filter frequency response and (b) its impulse response. Ideal filters : allow distortionless transmission of a certain band of frequencies and suppress all the remaining frequencies.

32. 32 Fundamental of Communication Systems ELCT332 Fall2011 Ideal high-pass and bandpass filter frequency responses. Paley-Wiener criterion

33. 33 Fundamental of Communication Systems ELCT332 Fall2011 Approximate realization of an ideal low-pass filter by truncating its impulse response. For a physically realizable system h(t) must be causal h(t)=0 for t<0

34. 34 Fundamental of Communication Systems ELCT332 Fall2011 Butterworth filter characteristics. The half-power bandwidth The bandwidth over which the amplitude response remains constant within 3dB. cut-off frequency

35. 35 Fundamental of Communication Systems ELCT332 Fall2011 Basic diagram of a digital filter in practical applications. Digital Filters Sampling, quantizing, and coding

36. 36 Fundamental of Communication Systems ELCT332 Fall2011 Pulse is dispersed when it passes through a system that is not distortionless. Linear Distortion Magnitude distortion Phase Distortion: Spreading/dispersion

37. 37 Fundamental of Communication Systems ELCT332 Fall2011 Signal distortion caused by nonlinear operation: (a) desired (input) signal spectrum; (b) spectrum of the unwanted signal (distortion) in the received signal; (c) spectrum of the received signal; (d) spectrum of the received signal after low-pass filtering. Distortion Caused by Channel Nonlinearities

38. 38 Fundamental of Communication Systems ELCT332 Fall2011 Multipath transmission. Multipath Effects

39. 39 Fundamental of Communication Systems ELCT332 Fall2011 Interpretation of the energy spectral density of a signal. Signal Energy: Parsevals Theorem Energy Spectral Density

40. 40 Fundamental of Communication Systems ELCT332 Fall2011 Figure 3.39 Estimating the essential bandwidth of a signal. Essential Bandwidth: the energy content of the components of frequeicies greater than B Hz is negligible.

41. 41 Fundamental of Communication Systems ELCT332 Fall2011 Find the essential bandwidth where it contains at least 90% of the pulse energy.

42. 42 Fundamental of Communication Systems ELCT332 Fall2011 Energy spectral densities of (a) modulating and (b) modulated signals. Energy of Modulated Signals

43. 43 Fundamental of Communication Systems ELCT332 Fall2011 Figure 3.42 Computation of the time autocorrelation function. Autocorrelation Function Determine the ESD of

44. 44 Fundamental of Communication Systems ELCT332 Fall2011 Limiting process in derivation of PSD. Signal Power Power Spectral Density Time Autocorrelation Function of Power Signals PSD of Modulated Signals