EcE 5013 Digital Signal Processing

1. EcE 5013 Digital Signal Processing Daw Mya Mya Aye Deputy Professor Department of Electronic Engineering & Information Technology Yangon Technological University

2. Overview • Continuous or analog signals • Discrete-time signals • Basic analog signals • Basic discrete signals • Quantized signals • Digital signals • Causal signals • Random signals • Representation of signals in MATLAB • Definition of a system • Block diagram

3. What is a signal? • A signal is a physical quantity, or quality, which conveys information • Example: voice of my friend is a signal which causes me to perform certain actions or react in a particular way • My friend's voice is called an excitation • My action or reaction is called a response

4. What is signal processing? • The conversion from excitation to response is called signal processing • A typical reason for signal processing is to eliminate or reduce an undesirable signal • We convert the original signal into a form that is suitable for further processing • One fundamental representation of a signal is as a function of at least one independent variable

5. What is a waveform • The variation of the signal value as a function of the independent variable is called a waveform • The independent variable often represents time • We define a signal as a function of one independent variable that contains information about the behavior or nature of a phenomenon • We assume that the independent variable is time even in cases where the independent variable is a physical quantity other than time

6. Continuous or analog signals • Continuous signal is a signal that exists at every instant of time • A continuous signal is often referred to as continuous time or analog • The independent variable is a continuous variable • Continuous signal can assume any value over a continuous range of numbers

7. Discrete-time signals • A signal defined only for discrete values of time is called a discrete-time signal or simply a discrete signal • Discrete signal can be obtained by taking samples of an analog signal at discrete instants of time • Digital signal is a discrete-time signal whose values are represented by digits

8. What is sampling? • Sampling is capturing a signal at an instant in time • Sampling means taking amplitude values of the signal at certain time instances • Uniform sampling is sampling every T units of time Sampling frequency or sampling rate time step or sample interval

9. Sinusoidal signal Phase in radian (rad) Amplitude Time in seconds (s) Frequency in Hertz (Hz)

10. MATLAB code for sine signal Xs = 1.8; fs = 10; fi = pi/3; t1 = -0.1; tstep = 0.01; t2 = 0.2; t = t1:tstep:t2; x = Xs*sin(2*pi*fs*t+fi); plot(t, x) xlabel('t') ylabel('x_s') title('x_s(t) = X_s sin(2 \pi f_s t + \phi_s)') grid on

11. Advanced MATLAB code Xs = 1.8; fs = 10; fi = pi/3; t1 = -0.1; t2 = 0.2; t = [t1, t2]; x = inline('Xs*sin(2*pi*fs*t+fi)','t','Xs','fs','fi'); fplot(x,t,2e-3,1,'-',Xs,fs,fi) xlabel('t'); ylabel('x_s'); grid on title('x_s(t) = X_s sin(2 \pi f_s t + \phi_s)')

12. Exponential signal x = inline('Xe*exp(b*t)','t','Xe','b'); Xe = 0.8; b = -0.5; t1 = 0; t2 = 8; t = [t1, t2]; fplot(x,t,2e-3,1,'-',Xe,b) xlabel('t') ylabel('x_e') title('x_e(t) = X_e e^{b t}') grid on

13. Unit step signal x = inline('t>0', 't'); t1 = -2; t2 = 6; t = [t1, t2]; fplot(x, t) xlabel('t') ylabel('u') title('Unit step signal') axis([t -0.1 1.1])

14. Pulse signal x = inline('(1/e)*((t>0)&(t<=e))','t','e'); e = 1/100; t1 = -1; t2 = 5; t = [t1, t2]; fplot(x,t,1e-5,1000,'-',e) set(gca,'FontSize',16) xlabel('t') ylabel('p_\epsilon(t)') axis([t -0.1 1.1/e]) title('Pulse function, \epsilon = 1/100')

15. Unit impulse signal (Dirac delta)

16. Causal signals • A signal is causal if it is zero for t < 0 • Causal signals are readily created by multiplying any continuous signal by the unit step signal • The instant when the signal begins is called the starting time • We usually take the starting time to be zero

17. Causal signals in MATLAB B = 0.02; a = 0.1; f = 0.53; phi = 3*pi/4; t = -5:0.05:10; x = B^a*exp(-a*t).*sin(... 2*pi*f*t+phi); xu = x.*(t>0); u = (t>0); plot(t,x,t,u,t,xu) ylabel('x(t)') xlabel('t (s)') text(0,1.2,'u(t)') text(-4,-1.1,'x(t)') text(5,-.6,'x(t)*u(t)') axis([t(1) t(end) -1.5 1.5])

18. Discrete-time signal – Sequence • A sequence (discrete-time signal, discrete signal, data sequence, or sample set) is a collection of ordered samples • In practical applications we process finite-length sequences • The existing sequence is often a sampled version of a continuous signal

19. Sinusoidal sequence Phase in radian (rad) Amplitude Sample index Period

20. Exponential sequence Xe = 0.8; a = 0.75; k1 = 0; k2 = 10; k = k1:k2; x = Xe*a.^k; stem(k, x) xlabel('k') ylabel('x_e') title('x_{e,k} = X_e a^k')

21. Unit step sequence k1 = -5; k2 = 10; k = k1:k2; x = (k>=0); stem(k, x) xlabel('k') ylabel('u_k') title('Unit step sequence') axis([k1 k2 -0.1 1.1])

22. Unit impulse sequence k1 = -5; k2 = 10; k = k1:k2; x = (k==0); stem(k, x) xlabel('k') ylabel('\delta_k') title('Unit impulse sequence') axis([k1 k2 -0.1 1.1])

23. Causal sequence • A sequence that is nonzero only over a finite interval of indices is called a finite-length sequence • A sequence whose samples are zero-valued for negative indices is causal • Anti-causal sequence can have nonzero samples only for negative indices

24. Causal sequences in MATLAB B = 0.94; a = 0.1; f = 0.32; k = -10:1:20; x = (B.^k).*cos(2*pi*f*k); u = (k>=0); ux = x.*u; subplot(3,1,1) stem(k,u,'g') ylabel('u_k','FontSize',14) axis([k(1) k(end) -2 2]) subplot(3,1,2) stem(k,x,'b') ylabel('x_k','FontSize',14) subplot(3,1,3) stem(k,ux,'r') ylabel('x_k u_k','FontSize',14) xlabel('k') axis([k(1) k(end) -2 2])

25. Quantized signal • The purpose of sampling a continuous signal is to transmit, store, or process a limited number of samples that are represented by a limited number of digits • By using fewer digits we attain faster transmission and smaller storage requirements for the information • We utilize the quantized samples rather than the true samples of infinite accuracy

26. Choice of digits for quantization • Choice of digits for quantization should be done properly: in transmitting, storing, and processing we prefer less digits • With too small a number of digits we can lose information from the original signal • Two opposing requirements must be satisfied: • Minimize number of digits to facilitate the signal transmission or storing, and • Maximize number of digits to keep the quantization error as low as necessary in order to preserve the information

27. Digital signal in MATLAB t = 0:30; x = 0.2+2*sin(0.245*t+0.15); d = 0.5; xq = d*round(x/d); plot(t,x) hold on stem(t,xq,'r') hold off ylabel('x(t), x_q(kT)') xlabel('t') legend('analog signal',... 'digital (quantized)')

28. Deterministic and random signal • Signal that can be described by an explicit mathematical form is deterministic • Deterministic signal can be periodic or aperiodic • Periodic signal consists of a basic shape of finite duration that is replicated infinitely • Signal that cannot be described in an explicit mathematical form is called random, also known as nondeterministic or stochastic

29. Random signal in MATLAB xk = 0:1:50; x = rand(size(k)); m = mean(x); s = std(x); stem(k,x) hold on plot([k(1) k(end)],… [m m],'r',... [k(1) k(end)],… [s s],'g') hold off xlabel('k') ylabel('x_k') ytick = [0 s m 1]; set(gca,'YTick',ytick) legend('random seq',… 'mean','std') title('Uniformly distributed samples')

30. Random signal in MATLAB k = 0:1:50; x = randn(size(k)); m = mean(x); s = std(x); stem(k,x) hold on plot([k(1) k(end)],… [m m],'r',… [k(1) k(end)],… [s s],'g') hold off xlabel('k') ylabel('x_k') legend('random seq',… 'mean','std') ytick = sort([-2 s m 2]); set(gca,'YTick',ytick) title('Normally distributed samples')

31. What is a system? • A signal is a physical quantity, or quality, which conveys information • Systems take one or more signals as input, perform operations on the signals, and produce one or more signals as output • A system is a group of related parts working together, or an ordered set of ideas, methods, or ways of working

32. Definition of a system • Implementation point-of-view: a system is an arrangement of physical components connected or related in such a manner as to form and/or act as an entire unit • Signal processing perspective: a system can be viewed as any process that results in the transformation of signals, in which systems act on signals in prescribed ways • Mathematical: a system as a mapping of N input signals onto M output signals; the mapping carries out a transformation on the input signals according to a set of rules

33. Basic definitions • Single-variable system (SISO system) has only one input and only one output • Multivariable system (MIMO system) has more than one input or more than one output • Input-output relationship (external description) is an equation that describes the relation between the input and the output of a system • Black box concept: the knowledge of the internal structure of a system is unavailable; the only access to the system is by means of the input ports and the output ports

34. Time response • One-dimensional system: required for processing a signal that is a function of the single independent variable • We assume that the independent variable is time even in cases where the independent variable is a physical quantity other than time • Time response is the output signal as a function of time, following the application of a set of prescribed input signals, under specified operating conditions

35. Continuous-time system Continuous-time system: the input and output signals are continuous time

36. Discrete-time system Discrete-time system has discrete-time input and output signals

37. Digital system • A discrete-time system is digital if it operates on discrete-time signals whose amplitudes are quantized • Quantization maps each continuous amplitude level into a number • The digital system employs digital hardware • explicitly in the form of logic circuits • implicitly when the operations on the signals are executed by writing a computer program

38. Analysis and design • Analysis of a system is investigation of the properties and the behavior (response) of an existing system • Design of a system is the choice and arrangement of systems components to perform a specific task • Design by analysis is accomplished by modifying the characteristics of an existing system • Design by synthesis: we define the form of the system directly from its specifications

39. Block diagram • Block diagram is a pictorial representation of a system that provides a method for characterizing the relationships among the components • Single block with one input and one output is the simplest form of the block diagram • Interior of the rectangle representing the block contains (a) component name, (b) component description, or (c) the symbol for the mathematical operation to be performed on input to yield output • Arrows represent the direction of signal flow

40. Elements of block diagram Takeoff point Summing point

41. Interconnections of blocks Blocks connected in cascade Blocks connected in feedback Blocks connected in parallel

42. State • For some systems, the output at time t0 depends not only on the input applied at t0, but also on the input applied before t0 • The state is the information at t0 that, together with input for t ≥ t0, determines uniquely output for t ≥ t0 • Dynamical equation is the set of equations that describes unique relations between the input, output, and state

43. Relaxed system • A system is said to be relaxed at time t0 if the output for t ≥ t0 is solely and uniquely determined by the input for t ≥ t0 • If the concept of energy is applicable, the system is said to be relaxed at t0 if no energy is stored in the system at t0 • A system is said to be zero-input if the output for t ≥ t0 is solely and uniquely determined by the state

44. Causality and stability • A system is called causal if the output depends only on the present and past values of the input • Intuitively, a stable system is one that will remain at rest unless excited by an external source and will return to rest if all excitations are removed • A relaxed system is BIBO stable (bounded-input bounded-output) if every bounded input produces a bounded output

45. Time-invariant system • A relaxed system is time-invariant if a time shift in the input signal causes a time shift in the output signal • In the case of discrete-time digital systems, we often use the term shift-invariant instead of time-invariant • Characteristics and parameters of a time-invariant system do not change with time

46. Linear system • Consider a relaxed system in which there is one independent variable t • A linear system is a system which has the property that if • input x1(t) produces an output y1(t) and • input x2(t) produces an output y2(t), then • input c1x1(t) + c2x2(t) produces an output c1y1(t) + c2y2(t) for any x1(t), x2(t) and arbitrary constants c1 and c2

47. Principle of superposition • The response y(t) of a linear system due to several inputs x1(t), x2(t), … xN(t) acting simultaneously is equal to the sum of the responses of each input acting alone • If yi(t) is the response due to the input xi(t), then

48. Linear time-invariant (LTI) system Continuous-time system is LTI if its input-output relationship can be described by the ordinary linearconstant coefficientdifferential equation

49. continued Discrete-time system is LTI if its input-output relationship can be described by the linear constant coefficientsdifference equation

50. Response of an LTI system • Free response (zero-input response) is the solution of the differential equation when the input is zero • Forced response (zero-state response) is the solution of the differential equation when the state is zero • Total response is the sum of the free response and the forced response • Total response can be viewed, also, as the sum of the steady-state response and transient response • Steady-state response is that part of the total response which does not approach zero as time approaches infinity • Transient response is that part of the total response which approaches zero as time goes to infinity