Lecture-1

1. Lecture-1 Signals & Systems

2. Signal is a set of information or data which changes with time. • Signal processing is a method of extracting information from signal by carrying out algorithmic operations on the signal. It is dependent on the type of signal and the nature of information it carries.

3. Analog Signal Processing

4. Digital circuits are less sensitive to changes in component values, temperature, ageing and other external parameters. • Advantages: • Flexibility in reconfiguration of DSP operations • Accurate • Easy storage • Cost effective

5. Digital Signal Processing

6. Applications of DSP

7. CLASSIFICATION OF SIGNALS • Continuous-Time and Discrete-Time Signals • Analog and Digital Signals • Real and Complex Signals • Deterministic and Random Signals • Even and Odd Signals • Periodic and aperiodic Signals • Energy and Power Signals • Causal, non-causal and anti-causal signals

8. Continuous-Time and Discrete-Time Signals • A signal x(t) is a continuous-time signal if t is a continuous variable. • If t is a discrete variable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal.

10. Analog and Digital Signals • An analog signal is a signal whose amplitude can take an infinite number of values. • A digital signal is a signal whose amplitude can take on a finite number of values. An analog signal may be converted into digital signal through sampling and quantization (rounding off). Sampling an analog signals results in the sampled analog signal which can still take on any value in a continuous range. It is digitized by rounding off its value to one of the closest permissible numbers (or quantized levels).

11. The amplitudes of the analog signal x(t) lie in the range of (-V, V). This range is partitioned into L-subintervals, each of magnitude: • Each sample amplitude is approximated by the midpoint value of the subinterval in which the sample falls. Each samples are approximated to one of the L-numbers. Thus, the signal is digitized with quantized samples taking on any one of the L-values. Such a signal is known as L-ary signal.

12. Real and Complex Signals • A signal x(t) is a real signal if its value is a real number. • A signal x(t) is a complex signal if its value is a complex number.

13. Deterministic and Random Signals • A signal whose physical description is known completely, either in a mathematical form or a graphical form is a deterministic signal. • A signal whose values cannot be predicted precisely but are known only in terms of probabilistic description, such as mean, mean squared value and so on is a random signal.

14. Even and Odd Signals

17. Periodic and aperiodic Signals • A signal, x(t), is said to be periodic if for some positive constant, T0: x(t)= x(t+ T0); for all t Smallest value of T0 that satisfies the above condition is called period of x(t). • A signal is aperiodic, if it does not follows the above equation.

19. Energy and Power Signals • A signal with finite energy is an energy signal. • A signal with finite and non-zero power is a power signal.

20. Energy Signal Power signal Total normalized power is finite and non-zero Periodic signals are power signals These signals can exist over infinite time Energy of the power signal is infinite • Total normalized energy is finite and non-zero. • Aperiodic signals are energy signals • These signals are time-limited • Power of energy signal is zero

21. Causal, non-causal and anti-causal signals • A signal that does not start before t=0 is a causal signal. In other words, x(t) is a causal if: x(t)=0; for t< 0 • A signal that starts before t=0 is a non-causal signal. • A signal that is zero for all t≥0 is called an anti-causal signal.

22. Useful signal operations • Time shifting • Time scaling • Time inversion (reversal) • Combined operations

26. Combined operations All the three operations are carried out simultaneously  x(at-b) • x(t) is time shifted by ‘b’: x(t-b) time scaling by ‘a’: x(at-b) • Time scale x(t) by ‘a’: x(at) time shift x(at) by ‘b/a’ (i.e. replace ‘t’ with (t-b/a))  x[a(t-b/a) =x(at-b) If ‘a’ is negative: Time scaling involves time inversion

27. BASIC CONTINUOUS-TIME SIGNALS • Unit Step Function • Unit Impulse Function • Complex Exponential Signals • Sinusoidal Signals

28. Unit Step Function

29. Unit Impulse Function

31. Sampling Property

33. Sifting Property

35. Complex Exponential Signals

37. Sinusoidal Signals

39. BASIC DISCRETE-TIME SIGNALS • Unit Step Sequence • Unit Impulse Sequence • Complex Exponential Sequences • Sinusoidal Sequences

40. Unit Step Sequence

41. Unit Impulse Sequence

43. Complex Exponential Sequences

45. Sinusoidal Sequences

47. SYSTEMS A system is a mathematical model of a physical process that relates the input (or excitation) signal to the output (or response) signal.

48. CLASSIFICATION OF SYSTEMS • Continuous-Time and Discrete-Time Systems • Systems with Memory and without Memory • Causal and Noncausal Systems • Linear Systems and Nonlinear Systems • Time-Invariant and Time-Varying Systems • Linear Time-Invariant Systems • Stable Systems • Feedback Systems

49. Continuous-Time and Discrete-Time Systems

50. Systems with Memory and without Memory A system is said to be memoryless if the output at any time depends on only the input at that same time. Otherwise, the system is said to have memory. Memoryless system Memory system