Digital signal processing lectures on signals

Electrical and electronics engineering
Digital signal processing

1. National Institute of Technology Calicut Digital Signal Processing Dr. Rakesh R Warier

2. Applications of Digital Signal Processing Audio and speech processing, Sonar, radar and other sensor array processing, Spectral density estimation, Digital image processing, Data compression, Signal processing for telecommunications, Control systems, Biomedical engineering, and Seismology, among others National Institute of Technology Calicut 2

3. Contents Review of Signals and Systems   Analysis and representation of discrete-time signal systems, including discrete-time convolution, Difference equations, the z-transform, and the discrete-time Fourier transform. Emphasis is placed on the similarities and distinctions between discrete-time, D igital filters, (FIR and IIR), Fast Fourier transform algorithm. National Institute of Technology Calicut 3

4. The most important algorithm! National Institute of Technology Calicut 4

5. Review of signals and systems What is a signal? What is a system? Analog vs Digital National Institute of Technology Calicut 5

6. Signals Continuous vs Discrete Signals A signal x(t) is analog and continuous-time if both x and t are continuous variables (infinite resolution). Most real world signals are analog and continuous-time . Dependent vs independent variable National Institute of Technology Calicut 6

7. Examples Continued National Institute of Technology Calicut 7 12 lead ECG signal

8. Discrete Signals Representations Bracket notation Stem plot   National Institute of Technology Calicut 8

9. Digital vs Discrete-time A signal x[n] is analog and discrete-time if the values of x are continuous but time n is discrete (integer-valued ). A signal x[n] is digital and discrete-time if the values of x are discrete (i.e., quantized) and time n also is discrete (integer-valued). Computers store and process digital discrete-time signals. Not covered in this course. National Institute of Technology Calicut 9

10. Deterministic vs Random Signals Any signal that can be represented by a unique model (formula, table or mathematical expression) without any uncertainty is called deterministic signal. In many practical applications, there are signals that cannot be described by any reasonable accuracy by any explicit mathematical formula, and they evolve in an unpredictable manner and are called random signals. National Institute of Technology Calicut 10

11. Continuous time Sinusoids National Institute of Technology Calicut 11 Continuous time sinusoids with different frequencies are distinct. Increasing frequency results in increasing oscillations

12. Periodic Signals (continuous) A periodic signal of period satisfies the periodicity property: for all integer values of n and all times t.   National Institute of Technology Calicut 12

13. Frequency (discrete time signal) A discrete signal is periodic with period N>0 if , for all Smallest value for which this is true is called the fundamental period.   National Institute of Technology Calicut 13

14. Example Find the frequency ( =50) Solve for N   National Institute of Technology Calicut 14

15. Example Find the frequency ( ) Solve for N   National Institute of Technology Calicut 15

16. Example Find the frequency ( ) Solve for N   National Institute of Technology Calicut 16 A discrete-time sinusoid is periodic only if the frequency is a rational number!

17. Frequency in discrete-time signals National Institute of Technology Calicut 17

18. Waveform Properties - Symmetry A signal x(t) exhibits even symmetry if its waveform is symmetrical with respect to the vertical axis . The shape of the waveform on the left-hand side of the vertical axis is the mirror image of the waveform on the right-hand side National Institute of Technology Calicut 18

19. Waveform Properties – Odd Symmetry National Institute of Technology Calicut 19 Odd Symmetry A signal exhibits odd symmetry if the shape of its waveform on the left-hand side of the vertical axis is the inverted mirror image of the waveform on the right-hand side.

20. Splitting Odd and Even Components National Institute of Technology Calicut 20

21. Fourier Formula National Institute of Technology Calicut 21

22. Fourier Series National Institute of Technology Calicut 22

23. Time – Frequency Duality National Institute of Technology Calicut 23

24. Fourier Series National Institute of Technology Calicut 24

25. Sampling National Institute of Technology Calicut 25 C onsider a sinusoidal signal x(t ) = cos ( 2 π 1000 t) , sampled at f s = 8000 samples per second. The sampling interval is T s = 1 / 8000 s,

26. Sampling Theorem Let x(t) be a real-valued, continuous-time, low pass signal bandlimited to a maximum frequency of B Hz. Let x [ n ] = x( nT s ) be the sequence of numbers obtained by sampling x(t) at a sampling rate of f s samples per second, that is, every T s = 1 /f s seconds. Then x(t) can be uniquely reconstructed from its samples x [ n ] if and only if f s > 2 B . The sampling rate must exceed double the bandwidth . The minimum sampling rate 2 B is called the Nyquist sampling rate . National Institute of Technology Calicut 26

27. National Institute of Technology Calicut 27

28. National Institute of Technology Calicut 28

29. Aliasing National Institute of Technology Calicut 29 The sampling rate must exceed double the bandwidth . The minimum sampling rate 2 B is called the Nyquist sampling rate .

30. Practice Problem National Institute of Technology Calicut 30

31. Solution National Institute of Technology Calicut 31

32. Sampling at Nyquist rate National Institute of Technology Calicut 32

33. Homework National Institute of Technology Calicut 33

34. References Digital Signal Processing: Principles, Algorithms, and Applications, John G .  Proakis . Northeastern University. Dimitris G. Manolakis . Boston College, 4 th ed. National Institute of Technology Calicut 34