ECE 3TR4 Communication Systems (Winter 2004)

1. ECE 3TR4 Communication Systems (Winter 2004) Dr. T. Kirubarajan (Kiruba) ECE Department CRL-225 kiruba@mcmaster.ca www.ece.mcmaster.ca/~kiruba/3tr4/3tr4.html

2. Course Overview • Communication Systems Overview • Fourier Series/Transform Review • Signals and Systems Review • Introduction to Noise • Motivation for Modulation • Amplitude Modulation • Angle Modulation • Pulse Modulation • Multiplexing • Transmitters and Receivers

3. Communication Systems Overview

4. Communication Systems Information Source Transmitter Channel Receiver Information Destination Blackberry Keypad Speakers Brain IP Packet GSM-style RF Vocal Tract SONET Router Wireless RF Acoustic Fiber FM Detector Ears Photo Diode ATM.25 Packet Brain Router  POTS Analog Communications (3TR4): Information is encoded in a continuous amplitude, continuous time signal. Digital Communications (4TK4): Information is encoded into a discrete time sequence with a quantized alphabet.

5. Communication Channels Channel: The medium linking the transmitter and receiver. It is ALWAYS analog in nature. That is every communication system is more or less ANALOG. • Channel Types • Wireline Channels: use a conductive medium to direct transmitted energy to the receiver: • Copper wire for telephones, xDSL • Fiber optic cable • Aluminum interconnects for ICs • Wireless Channels: Uses an open propagation medium • RF for cell phones • Underwater acoustic ducts for whales

6. Channel Impairments As a transmitted signal propagates it loses fidelity in a number of ways. This loss of fidelity makes the received signal look very different from the transmitted signal. Additive Noise: Thermal noise, multi-transmitter interference Noise + Transmitter Receiver Multiplicative Noise: Rayleigh Fading Noise x Transmitter Receiver Convolution Noise: time-delay multipath, reverberation Noise Transmitter Receiver

7. 3TR4 Objective Information Source Transmitter Channel Receiver Information Destination 1. How to design 2. Taking into account • 3. That will provide a system that is: • Reliable: information received is what was sent • Efficient: Not wasteful of time, power or spectrum • Simple: economical for H/W and S/W and usually Robust

8. Tradeoffs in Objectives Spectral Use Efficient Temporal Use Power Use Simple Reliable Simple H/W Accuracy & Robustness Simple S/W

9. Digital Communications N Digital Information Source Source Encoder Channel Encoder Modulator DAC The placement of the DAC and ADC is up to the system requirements. They can be anywhere between the Information Sources and Destination and the Modulator and Demodulator, respectively. Channel Digital Information Destination Source Decoder Channel Decoder DeModulator ADC

10. Fourier Series/Transform Review

11. Fourier Review Fourier Series and Transforms try to form a signal out of sinusoids. These sinusoids have a specific frequency and go on forever. That is your nice time series which is represented by points in time will now be represented by points in frequency. This is why we use the terms “Fourier domain” and “frequency domain” interchangeably. Reminder:

12. What Transform, When?

13. Discrete Time Fourier Series DTFS: I-DTFS: X[k] and x[n] have period N Ω0 = 2π/N

14. Discrete Time Fourier Transform DTFT: I-DTFS: X[k] has period 2π

15. Fourier Series FS: I-FS: X(t) has period T Ω0 = 2π/T

16. Fourier Transform FT: I-FT: The Fourier Transform is the general transform, it can handle periodic and non-periodic signals. For a periodic signal it can be thought of as a transformation of the Fourier Series

17. Fourier Series

18. Fourier Series – Real Signals If x(t) is real valued: Ak = A-k Bk = -B-k

19. Fourier Series – Real +Even/Odd Even: f(t) = f(-t), therefore Bk = 0; Cosine Series Odd: f(t) = -f(-t), therefore Ak = 0; Sine Series

20. Cosine Fourier Series Even Function FS FT = 2π(FS) When is FT the continuous counterpart to 2πFS? How do the Delta’s move as frequency changes?

21. Sine Fourier Transform Odd Function FS FT = 2π(FS) The Fourier Transform of an Odd Signal is Odd. Notice the Fourier Domain graph is in jF(ω). It is imaginary.

22. DC Fourier Transform DC Function FS FT (FS) FT The FT of a signal with a DC component is separable. The DC component of a time signal is statistically the MEAN.

23. Delta Fourier Transform Delta Function FS - No Fourier Series, Not Periodic FT The FT is only congruent with the FS for PERIODIC signals. A delta has an infinitely steep rise time, therefore it has a great deal of high frequencies

24. Pulse Train Fourier Transform Function with Period T FS What happens in the Frequency Domain when the time between pulses is shortened? When T  0? When T = 0?

25. Time Window Fourier Transform Not Periodic – No FS FT

26. Ideal Filter Fourier Transform Not Periodic – No FS FT Why is this called the “ideal filter”? Notice similarities between this and rectangular time window, and how W here is a counterpart to τ there in controlling width.

27. Triangle Fourier Transform Not Periodic – No FS FT Sinc squared can never be negative. Why are we introducing these signals? They are the foundation of most analog communication signals.

28. More Complex Example An pulse train with period (T) one second is convolved with a time windowing function with timing (τ) of 0.5 seconds, to produce a 50% duty cycle square wave.

29. More Complex Example The spectrum of the pulse train is: The spectrum of the square-wave is: Convolution turns into Multiplication in the Freq Domain This turns into a line spectra, and how it changes with changing the parameters is very informative

30. Constant τ T = 2 T = 4 T = 8 • Amplitude DECREASES as 1/T • Line spectra resolution INCREASES as T • The envelope is INDEPENDENT of T

31. Constant T T = 2 • Amplitude INCREASES in proportion to Tau • Line spectra resolution is INDEPENDENT of Tau • The spectrum SPREADS as the window shortens • !!! TIME RESOLUTION AND FREQUENCY RESOLUTION ARE INVERSELY RELATED !!!!!!!!

32. The Sampling Theorem One of the fundamental concepts in dealing with the representation of analog signals in the digital domain is the Nyquist Rate, or Minimum Time-Bandwidth product. This law states the minimum sample frequency necessary to exactly represent an analog signal as a digital signal. Since one of the main constraints in judging the efficiency of a communication system is spectral efficiency, the Nyquist rate forms a large part of the back-bone of system design. A real-valued band-limited signal having no spectral components above a frequency of B Hz is determined uniquely by its values at uniform intervals spaced no greater than 1/2B seconds apart

33. Sampling Theorem Consider a signal f(t) sampled with an impulse train p(t)

34. Sampling Theorem Visual Band limited signal + spectrum Periodic gating function + spectrum Size of sampling window controls envelope of spectrum, sample frequency controls spacing of original spectrum replicas

35. Nyquist Rate Since the periodic gating function controls the center of the replicas and the replicas are 2W (W = 2πB) wide, then to make sure there is no overlap: If the signal is sampled at a lower rate there will be overlap, and in the final spectrum you won’t know if the overlapped part is from the spectrum that is suppose to be there or from the “ALIASED” part of the spectrum

36. Signals and Systems Review

37. Energy and Power Signal Energy Signal Power An energy signal cannot be a power signal, nor vice-versa To be an energy signal: Amplitude  0 As |Time|  inf

38. Energy and Power Example Find Ex 0

39. Parseval’s Theorem Energy calculated in the Time domain is equal to energy calculated in the Frequency domain.

40. Power Spectral Density

41. PSD Sf(ω) is the power spectral density function, it has units of power per Hz. Gf(ω) is the cumulative spectral power function, it the amount of energy in the signal in those components less then ω.

42. Autocorrelation

43. Autocorrelation Rf(τ) should look familiar in a way. It is equivalent to convolving the function f(t) with f(-t). The autocorrelation function is often used for signal detection in a background of random noise. When we get into random noise it will become very evident why this is so.

44. Linear Time Invariant Systems Fundamental way of describing many components in a communication system. Models filters, amplifiers and equalizers very well. Model an LTI system with the impulse response, h(t), of the system, the response of an impulse input to the system. The Fourier Transform of the impulse response is the frequency transfer function. x(t) h(t) y(t)

45. Time Operators f(t)  g(t)  f(t-a)  g(2t)  f(t+b)  g(t/2)  What happens to in the Fourier domain to each of these?

46. Invertibility LTI systems are invertible If you can determine the input given the output then a system is called Invertible Given input x and it’s output is y: y(t) = 2 x(t) Is inverted by z: z(t) = ½ y(t) = x(t) Not invertible: y(t) = floor{x(t)} !!! A non-invertible system usually maps multiple points from the input space to the same point in the output space.

47. LTI Systems x(t) h(t) y(t) H(ω) X(ω) Y(ω) In the frequency domain the convolution integral becomes a multiplication, and vice-versa. By assessing the frequency domain magnitude and phase we can see how H can effect specific frequencies differently: !!! This is the beginning of the filtering interpretation

48. LTI Systems The Law of Superposition: Given inputs a and b to system x, a linear system: x(a)+x(b) = x(a+b) Given input a and some scalar constant to system x, x(c a) = c x(a) The Law of Time Invariance: Given some input function g(t) and is input to a system X produces an output f(t) X{g(t)} = f(t) If g(t) is shifted in time by T0 then the output has the same shift X{g(t-T0)} = f(t-T0) The Law of Commutation: Given some function g(t) and f(t) g(t) * f(t) = f(t) * g(t)

49. Ideal Filter Introduction Frequency Response Impulse Response Low Pass Filter (LPF) High Pass Filter (HPF) BandPass Filter (BPF) BandStop Filter (BSF)

50. Real Filters In reality one cannot make the Brick Wall type ideal filters. This is due to the fundamental tradeoff between time and frequency resolution. If you have a jump in the frequency response that is infinitesimally resolved, you’d need infinite time to represent that. One deals with filter specifications such as bandwidth, roll-off, implementation complexity, passband ripple and so on for most of this course, and for many future courses. It is of great practical importance to understand the tradeoffs implicit in the time-frequency bandwidth tradeoff.