Signal and System Analysis

1. Signal and System Analysis • Objectives: • To define the energy and power of a signal • To classify different types of signals • To introduce special functions commonly used in telecom. • To define linear and time-invariant systems • To define convolution • To introduce Fourier series and Fourier transform • To explain the concept of negative frequency • To show how the signal may be described in either the time domain or the frequency domain and establish their relationship. • To study autocorrelation and power spectral density

2. Classification of Signals As mentioned before, a signal represents the message that is to be sent across the channel. Let’s start looking into more detail. • Signals can be classified in various ways: • Continuous- or discrete-time signals Signals associated with a computer are discrete-time signals

3. Classification of Signals Periodic and nonperiodic signals A periodic signal is one that repeats itself exactly after a fixed length of time. g(t) = g(t + T) for all t g[n] = g[n+N] for all n The smallest positive number T (or N) that satisfies the above equation is called the period.

4. Classification of Signals Deterministic and random signals Deterministic signal: can be mathematically characterizedcompletely in the time domain. Random signal: specified only in terms of probabilistic description All signals encountered in telecommunications are random signals. If a message is used to convey information, it must have some uncertainty (randomness) about it. Otherwise, why communicate?

5. Power and energy of a signal • We know ,power • Energy • For our purpose, we will neglect the • Energy and average power of a signal and

6. Some important signals • Singularity functions in continuous-time systems • Singularity functions are discontinuous or have discontinuous derivatives. • Singularity functions are mathematical idealizations and, strictly speaking, do not occur in physical systems. • They serve as good approximations to certain limiting conditions in physical systems. • We will discuss two types of singularity functions: • unit step function and • unit impulse function

7. Step function Unit step function The unit step function is defined as (u(t) has no definition at t = 0, or one may define u(0) = 1 or u(0) = ½)

8. Impulse function Unit impulse function The unit impulse function is defined as

9. Relationship between (t) and u(t)

10. Further properties of (t), [n] Multiplication of a function by (t) f(t)(t) = f(0)(t), f(t) continuous at t=0. f(t)(t – T) = f(T)(t – T),f(t) continuous at t=T. Sampling property of (t), [n]

11. Exponential functions Euler’s formula Exponential functions are important class of functions in this course. We will be making use of this fact a lot in this course

12. Systems A system is defined as a set of rules that maps an input signal to an output signal. g(t) -- input signal (or source signal); y(t) -- output signal (or response signal); Input and response are represented as g(t)  y(t) and read as input g(t) causes a response y(t).

13. Some properties of systems • Linear and nonlinear systems • For a system with • g1(t)  y1(t) and g2(t)  y2(t), • A system is said to be linear if the following properties hold: • ag1(t)  ay1(t) • ag1(t) + bg2(t)  ay1(t) +by2(t) for any scalar a,b • Otherwise, the system is nonlinear. • Examples

14. Some properties of systems Time-invariant and time-varying systems A system is time-invariant if a time shift in the input results in a corresponding time shift in the output g(t – t0)  y(t – t0) for any t0. i.e. when the same input is applied to a system today or tomorrow, the output is the same, just shifted in time accordingly

15. Exercise Time Invariance Any system not meeting this requirement is said to be time-varying. Example: Another example: Humans We will focus on Linear and Time-Invariant (LTI) systems in this course

16. Exercise Non-linear, time-invariant Non-linear, time-invariant Linear, time-variant Non-linear, time-invariant Linear, time-invariant

17. Convolution in LTI systems Consider a discrete-time LTI system. If we apply the impulse function as the input, let be the output. We call the Impulse response of an LTI system Now, since the system is linear, if we apply a scaled version of the impulse ,

18. Convolution in LTI systems Furthermore, since the system is time-invariant, if we apply a delayed version of the impulse , Now, what if we have this?

19. Convolution in LTI systems If we define Do you see a pattern? In an LTI system with input and impulse response , Similarly for continuous-time systems,

20. Convolution in LTI systems We write convolution as Example:

21. Convolution in LTI systems LTI system In an LTI system, the impulse response or completely characterize the system.

22. Amplitude Amplitude Fourier Analysis Fourier analysis Alternative Representation of a periodic signal A signal can be represented in either the time domain (where it is a waveform as a function of time) or in the frequency domain. Such representation is called the spectrum of the original time-domain signal. If the signal is specified in the time domain, we can determine its spectrum, and vice versa. Fourier analysis provides a link between the time domain and the frequency domain.

23. Fourier Analysis Claim: A periodic function g(t) of period T, can be expressed as an infinite sum of sinusoidal waveforms with frequency This summation is called Fourier series. Fourier series can be written in several forms. One such form is the trigonometric Fourier series: The constants are called Fourier Coefficients

24. Obtaining the Fourier Coefficients

25. Fundamental Frequency and Harmonics Given with period T, in the formulation is referred to as the fundamental frequency and the others are called harmonics of g(t)

26. Example

27. Fourier serious approximation to periodic functions In practice, As , the Fourier series approaches the original function

28. Sinc function Rewrite: If we define a function and define sinc(0)=1, we may rewrite This is called the sinc function

29. Sinc function

30. Exponential Fourier series 0 = 2/T Since and we can rewrite as where This is called the exponential Fourier series

31. Exponential Fourier series Exponential Fourier series How are the coefficients in the exponential Fourier series cn related to an , bn? where c0 = a0 / 2, cn = (an – jbn) / 2, c-n= (an + jbn) / 2

32. Fourier coefficients cn in complex fourier series Since c0 = a0 / 2, cn = (an – jbn) / 2, c-n= (an + jbn) / 2, cn can be complex. • | cn | is called spectral amplitude and represents the amplitude of nth harmonic. Arg(cn) is known as the spectral phase • Graphic representation of spectral amplitude along with the spectral phase is called complex frequency spectrum of the original signal g(t)

33. Negative Frequency ~!?!? Since n takes on negative values, apprently there is “negative frequency”. But remember, this exists only because we want to express and as In reality, frequencies can only have positive values. However, it is mathematically easier to use exponential representation rather than trigonometric. That’s why we allow the existence of negative frequency. Keep this in mind ~

34. A bit of revision Up to now, we have shown that a periodic function in time g(t)can be specified in two equivalent ways: Time domain representation--waveform. Frequency domain representation–spectrum, Fourier coefficients. If the signal is specified in time domain, we can determine its spectrum and vice versa.

35. Spectrum dependence on period of signal

36. Spectrum dependence on period of signal • When T is larger, becomes smaller, the spectrum becomes denser. • When T goes to infinity, • Only a single pulse of width  in the time domain. • 0 0, i.e., no spacing is left between two line-components; • Thus, the spectrum becomes continuous and exists at all frequencies. (However, there is no change in the shape of the envelope of the spectrum).

37. Fourier Transform The Fourier transform of a signal g(t) is defined by and g(t) is called the inverse Fourier transform of G() The functions g(t) and G() constitute a Fourier transform pair: g(t)  G() G() = F[g(t)] and g(t) = F -1[G()] What is the difference between Fourier transform and Fourier series?

38. Fourier Transform Fourier transform is different from the Fourier Series in that its frequency spectrum is continuous rather than discrete. Fourier transform is obtained from Fourier series by letting T  (for a nonperiodic signal). The original time function can be uniquely recovered from its Fourier transform.

39. Fourier Transform and Fourier Series • Please keep in mind that • A periodic signal spectrum has finite amplitudes and exists at discrete set of frequencies. Those amplitudes are also called the Fourier coefficients of the periodic signal • A non-periodic signal has a continuous spectrumG() and exist at all frequencies.

40. Fourier transform of some useful functions Rectangular function: Proof

41. Fourier transform of some useful functions Unit impulse function: (t)  1 and 1  2() Proof

42. Fourier transform of some useful functions Sinusoidal function cos(0t) cos(0t) [( + 0) + ( - 0)] Proof

43. Properties of Fourier Transform Linearity property If g1(t)  G1() and g2(t)  G2() then a1g1(t) + a2g2(t)  a1G1() + a2G2() where a1 and a2 are constants This property is proved easily by linearity property of integrals used in defining Fourier transform

44. Properties of Fourier Transform Symmetry property If g(t)  G(), then G(t)  2g(- ) Proof we can interchange the variable t and , i.e. let t ,  t, then

45. Properties of Fourier Transform Time scaling property Proof let x = at, then dt = dx/a, case 1: when a > 0, case 2: when a < 0, then t  leads to x  - , Combined, the two cases are expressed as,

46. Properties of Fourier Transform Important Observation: Time domain compression of a signal results in spectral expansion Time domain expansion of a signal results in spectral compression

47. Properties of Fourier Transform Time shifting property Proof put t – t0 = x, so that dt = dx, then Frequency shifting property Proof

48. Properties of Fourier Transform • Significance • Multiplication of a function g(t) by exp(j0t) is equivalent to shifting its Fourier transform in the positive direction by an amount 0. -- Frequency translation theorem. • Translation of a spectrum helps in achieving modulation, which is performed by multiplying the known signal g(t) by a sinusoidal signal. Therefore,

49. Modulation Theorem • The multiplication of a time function with a sinusoidal function translates the whole spectrum G() to 0. • exp(j0t) can also provide frequency translation, but it is not a real signal. Hence, sinusoidal function is used in practical modulation system.

50. Properties of Fourier Transform Convolution Suppose that g1(t)  G1() and g2(t)  G2(), then, what is the waveform of g(t) whose Fourier transform is the product of G1() and G2()? This question arises frequently in spectral analysis, and is answered by the convolution theorem. The convolution of two time function g1(t) and g2(t), is defined by the following integral