Review of DSP

1. Review of DSP

2. Signal and Systems: • Signal are represented mathematically as functions of one or more independent variables. • Digital signal processing deals with the transformation of signal that are discrete in both amplitude and time. • Discrete time signal are represented mathematically as sequence of numbers.

3. Signals and Systems: • A discrete time system is defined mathematically as a transformation or operator. • y[n] = T{ x[n] } T{.} x [n] y [n]

4. Linear Systems: • The class of linear systems is defined by the principle of superposition. • And • Where a is the arbitrary constant. • The first property is called the additivity property and the second is called the homogeneity or scaling property.

5. Linear Systems: • These two property can be combined into the principle of superposition, H Linear System H H

6. Time-Invariant Systems: • A Time-Invariant system is a system for which a time shift or delay of the input sequence cause a corresponding shift in the output sequence. H H

7. LTI Systems: • A particular important class of systems consists of those that are linear and time invariant. • LTI systems can be completely characterized by their impulse response. • From principle of superposition: • Property of TI:

8. LTI Systems (Convolution): • Above equation commonly called convolution sum and represented by the notation

9. Convolution properties: • Commutativity: • Associativity: • Distributivity: • Time reversal:

10. H1 H2 H2 H1 …Convolution properties: • If two systems are cascaded, • The overall impulse response of the combined system is the convolution of the individual IR: • The overall IR is independent of the order:

11. Duration of IR: • Infinite-duration impulse-response (IIR). • Finite-duration impulse-response (FIR) • In this case the IR can be read from the right-hand side of:

12. Transforms: • Transforms are a powerful tool for simplifying the analysis of signals and of linear systems. • Interesting transforms for us: • Linearity applies: • Convolution is replaced by simpler operation:

13. …Transforms: • Most commonly transforms that used in communications engineering are: • Laplace transforms (Continuous in Time & Frequency) • Continuous Fourier transforms (Continuous in Time) • Discrete Fourier transforms (Discrete in Time) • Z transforms (Discrete in Time)

14. The Z Transform: • Definition Equations: • Direct Z transform • The Region Of Convergence (ROC) plays an essential role.

15. The Z Transform (Elementary functions): • Elementary functions and their Z-transforms: • Unit impulse: • Delayed unit impulse:

16. The Z Transform (…Elementary functions): • Unit Step: • Exponential:

17. Z Transform (Cont’d) • Important Z Transforms Region Of Convergence (ROC) Whole Page Whole Page |z| > 1 |z| > |a|

18. The Z Transform (Elementary properties): • Elementary properties of the Z transforms: • Linearity: • Convolution: if ,Then

19. The Z Transform (…Elementary properties): • Shifting: • Differences: • Forward differences of a function, • Backward differences of a function,

20. The Z Transform (…Region Of Convergence for Z transform): • Since the shifting theorem

21. The Z Transform (Region Of Convergence for Z transform): • The ROC is a ring or disk in the z-plane centered at the origin :i.e., • The Fourier transform of x[n] converges at absolutely if and only if the ROC of the z-transform of x[n] includes the unit circle. • The ROC can not contain any poles.

22. The Z Transform (…Region Of Convergence for Z transform): • If x[n] is a finite-duration sequence, then the ROC is the entire z-plane, except possibly or . • If x[n] is a right-sided sequence, the ROC extends outward from the outermost finite pole in to . • The ROC must be a connected region.

23. The Z Transform (…Region Of Convergence for Z transform): • A two-sidedsequence is an infinite-duration sequence that is neither right sided nor left sided. • If x[n] is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and not containing any poles. • If x[n] is a left-sided sequence, the ROC extends in ward from the innermost nonzero pole in to .

24. The Z Transform (Application to LTI systems): • We have seen that • By the convolution property of the Z transform • Where H(z) is the transfer function of system. • Stability • A system is stable if a bounded input produced a bounded output, and a LTI system is stable if:

25. Fourier Transform Time Transform Type Frequency Continuous-aperiodic Continuous-aperiodic Fourier Transform Continuous-periodic Discrete-aperiodic DiscreteTime Continuous Frequency FT Continuous-periodic Discrete-aperiodic Fourier Series Discrete-periodic Discrete-periodic Discrete Time Discrete Frequency FT

26. Discrete-time Fourier Transform The same as Z-transform with z on the unit circle Continuous in Frequency, periodic with period = 2*pi

27. The Discrete Fourier Transform (DFT) • Discrete Fourier transform • It is customary to use the • Then the direct form is:

28. The Discrete Fourier Transform (DFT) • With the same notation the inverse DFT is

29. The DFT (Elementary functions): • Elementary functions and their DFT: • Unit impulse: • Shifted unit impulse:

30. The DFT (…Elementary functions): • Constant: • Complex exponential:

31. The DFT (…Elementary functions): • Cosine function:

32. The DFT (Elementary properties): • Elementary properties of the DFT: • Symmetry: If ,Then • Linearity: if and ,Then

33. The DFT (…Elementary properties): • Shifting: because of the cyclic nature of DFT domains, shifting becomes a rotation. if ,Then • Time reversal: if ,Then

34. The DFT (…Elementary properties): • Cyclic convolution: convolution is a shift, multiply and add operation. Since all shifts in the DFT are circular, convolution is defined with this circularity included.