EC 2314 Digital Signal Processing

1. EC 2314 Digital Signal Processing By Dr. K. Udhayakumar

2. Signal • A signal is a pattern of variation that carry information. • Signals are represented mathematically as a function of one or more independent variable • A picture is brightness as a function of two spatial variables, x and y. • In this course signals involving a single independent variable, generally refer to as a time, t are considered. Although it may not represent time in specific application • A signal is a real-valued or scalar-valued function of an independent variable t.

3. Signal Types Signals Continuous-time Discrete-time Continuous-value Discrete-value Continuous-value Discrete Digital Analog

4. Signal Types • Analog signals: continuous in time and amplitude • Example: voltage, current, temperature,… • Digital signals: discrete both in time and amplitude • Example: attendance of this class, digitizes analog signals,… • Discrete-time signals: discrete in time, continuous in amplitude • Example: hourly change of temperature • Theory of digital signals would be too complicated • Requires inclusion of nonlinearities into theory • Theory is based on discrete-time continuous-amplitude signals • Most convenient to develop theory • Good enough approximation to practice with some care • In practice we mostly process digital signals on processors • Need to take into account finite precision effects

5. Signal Types • Continuous time – Continuous amplitude • Continuous time – Discrete amplitude • Discrete time – Continuous amplitude • Discrete time – Discrete amplitude

6. Example of signals • Electrical signals like voltages, current and EM field in circuit • Acoustic signals like audio or speech signals (analog or digital) • Video signals like intensity variation in an image • Biological signal like sequence of bases in gene • Noise which will be treated as unwanted signal

7. Signal classification • Continuous-time and Discrete-time • Energy and Power • Real and Complex • Periodic and Non-periodic • Analog and Digital • Even and Odd • Deterministic and Random

8. A continuous-time signal • Continuous-time signal x(t), the independent variable, t is Continuous-time. The signal itself needs not to be continuous.

9. x(t) t Continuous Time (CT) Signals • Most signals in the real world are continuous time, as the scale is infinitesimally fine. • E.g. voltage, velocity, • Denote by x(t), where the time interval may be bounded (finite) or infinite

10. A piecewise continuous-time signal • A piecewise continuous-time signal

11. x[n] n Discrete Time (DT) Signals • Some real world and many digital signals are discrete time, as they are sampled • E.g. pixels, daily stock price (anything that a digital computer processes) • Denote by x[n], where n is an integer value that varies discretely • Sampled continuous signal • x[n] =x(nk)

12. A discrete-time signal • A discrete signal is defined only at discrete instances. Thus, the independent variable has discrete values only.

13. Sampling • A discrete signal can be derived from a continuous-time signal by sampling it at a uniform rate. • If denotes the sampling period and denotes an integer that may assume positive and negative values, • Sampling a continuous-time signal x(t) at time yields a sample of value • For convenience, a discrete-time signal is represented by a sequence of numbers: • We write • Such a sequence of numbers is referred to as a time series.

14. 2p Periodic Signals • An important class of signals is the class of periodic signals. A periodic signal is a continuous time signal x(t), that has the property • where T>0, for all t. • Examples: • cos(t+2p) = cos(t) • sin(t+2p) = sin(t) • Are both periodic with period 2p

15. Odd and Even Signals • An even signal is identical to its time reversed signal, i.e. it can be reflected in the origin and is equal to the original: • Examples: x(t) = cos(t) x(t) = c • An odd signal is identical to its negated, time reversed signal, i.e. it is equal to the negative reflected signal • Examples: x(t) = sin(t) x(t) = t • This is important because any signal can be expressed as the sum of an odd signal and an even signal.

16. Exponential and Sinusoidal Signals • Exponential and sinusoidal signals are characteristic of real-world signals and also from a basis (a building block) for other signals. • A generic complex exponential signal is of the form: • where C and a are, in general, complex numbers. Lets investigate some special cases of this signal • Real exponential signals Exponential growth Exponential decay

17. Periodic Complex Exponential & Sinusoidal Signals • Consider when a is purely imaginary: • By Euler’s relationship, this can be expressed as: • This is a periodic signals because: • when T=2p/w0 • A closely related signal is the sinusoidal signal: • We can always use: cos(1) T0 = 2p/w0 = p T0 is the fundamental time period w0 is the fundamental frequency

18. Exponential & Sinusoidal Signal Properties • Periodic signals, in particular complex periodic and sinusoidal signals, have infinite total energy but finite average power. • Consider energy over one period: • Therefore: • Average power: • Useful to consider harmonic signals • Terminology is consistent with its use in music, where each frequency is an integer multiple of a fundamental frequency

19. General Complex Exponential Signals • So far, considered the real and periodic complex exponential • Now consider when C can be complex. Let us express C is polar form and a in rectangular form: • So • Using Euler’s relation • These are damped sinusoids

20. Discrete Unit Impulse and Step Signals • The discrete unit impulse signal is defined: • Useful as a basis for analyzing other signals • The discrete unit step signal is defined: • Note that the unit impulse is the first difference (derivative) of the step signal • Similarly, the unit step is the running sum (integral) of the unit impulse.

21. Continuous Unit Impulse and Step Signals • The continuous unit impulse signal is defined: • Note that it is discontinuous at t=0 • The arrow is used to denote area, rather than actual value • Again, useful for an infinite basis • The continuous unit step signal is defined:

22. A piecewise discrete-time signal • A piecewise discrete-time signal

23. Energy and Power Signals • X(t) is a continuous power signal if: • X[n] is a discrete power signal if: • X(t) is a continuous energy signal if: • X[n] is a discrete energy signal if:

24. Power and Energy in a Physical System • The instantaneous power • The total energy • The average power

25. Energy and Power over Infinite Time • For many signals, we’re interested in examining the power and energy over an infinite time interval (-∞, ∞). These quantities are therefore defined by: • If the sums or integrals do not converge, the energy of such a signal is infinite • Two important (sub)classes of signals • Finite total energy (and therefore zero average power) • Finite average power (and therefore infinite total energy) • Signal analysis over infinite time, all depends on the “tails” (limiting behaviour)

26. Power and Energy • By definition, the total energy over the time interval in a continuous-time signal is: denote the magnitude of the (possibly complex) number • The time average power • By definition, the total energy over the time interval in a discrete-time signal is: • The time average power

27. Power and Energy • Example 1: The signal is given below is energy or power signal. Explain. • This signal is energy signal

28. Power and Energy • Example 2: The signal is given below is energy or power signal. Explain. • This signal is energy signal

29. Real and Complex • A value of a complex signal is a complex number • The complex conjugate, of the signal is; • Magnitude or absolute value • Phase or angle

30. Periodic and Non-periodic • A signal or is a periodic signal if Here, and are fundamental period, which is the smallest positive values when Example:

31. Analog and Digital Digital signal is discrete-time signal whose values belong to a defined set of real numbers • Binary signal is digital signal whose values are 1 or 0 • Analog signal is a non-digital signal

32. Even and Odd • Even Signals The continuous-time signal /discrete-time signal is an even signal if it satisfies the condition • Even signals are symmetric about the vertical axis • Odd Signals The signal is said to be an odd signal if it satisfies the condition • Odd signals are anti-symmetric (asymmetric) about the time origin

33. Even and Odd signals:Facts • Product of 2 even or 2 odd signals is an even signal • Product of an even and an odd signal is an odd signal • Any signal (continuous and discrete) can be expressed as sum of an even and an odd signal:

34. Complex-Valued Signal Symmetry • For a complex-valued signal is said to be conjugate symmetric if it satisfies the condition where is the real part and is the imaginary part; is the square root of -1

35. Deterministic and Random signal • A signal is deterministic whose future values can be predicted accurately. • Example: • A signal is random whose future values can NOT be predicted with complete accuracy • Random signals whose future values can be statistically determined based on the past values are correlated signals. • Random signals whose future values can NOT be statistically determined from past values are uncorrelated signals and are more random than correlated signals.

36. Deterministic and Random signal(contd…) • Two ways to describe the randomness of the signal are: • Entropy: This is the natural meaning and mostly used in system performance measurement. • Correlation: This is useful in signal processing by directly using correlation functions.

37. Basic sequences and sequence operations • Delaying (Shifting) a sequence • Unit sample (impulse) sequence • Unit step sequence • Exponential sequences

38. Discrete-Time Systems • A Discrete-Time System is a mathematical operation that maps a given input sequence x[n] into an output sequence y[n] Example: Moving (Running) Average Maximum Ideal Delay System

39. Memoryless System A system is memoryless if the output y[n] at every value of n depends only on the input x[n] at the same value of n Example : Square Sign counter example: Ideal Delay System

40. Linear Systems • Linear System: A system is linear if and only if Example: Ideal Delay System

41. Time-Invariant Systems Time-Invariant (shift-invariant) Systems A time shift at the input causes corresponding time-shift at output Example: Square Counter Example: Compressor System

42. Causal System A system is causal iff it’s output is a function of only the current and previous samples Examples: Backward Difference Counter Example: Forward Difference

43. Stable System Stability (in the sense of bounded-input bounded-output BIBO). A system is stable iff every bounded input produces a bounded output Example: Square Counter Example: Log

44. LTI System Example

45. Linear Time-Invariant Systems • Special importance for their mathematical tractability • Most signal processing applications involve LTI systems • LTI system can be completely characterized by their impulse response

46. Digital Signal Processing

47. Ex)

49. Properties of LTI Systems • Convolution is commutative • Convolution is distributive

50. Properties of LTI Systems • Cascade connection of LTI systems