Review of DSP 1 Signal and Systems: Signal are represented - PowerPoint PPT Presentation

Review of DSP 1

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. 2

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] 3

Linear Systems:  The class of linear systems is defined by the principle of superposition.      { [ ] [ ]} { [ ]} { [ ]} [ ] [ ] T x n x n T x n T x n y n y n 1 2 1 2 1 2  And   { [ ]} { [ ]} [ ] T ax n aT x n ay n  Where a is the arbitrary constant.  The first property is called the additivity property and the second is called the homogeneity or scaling property. 4

Linear Systems:  These two property can be combined into the principle of superposition,    { [ ] [ ]} { [ ]} { [ ]} T ax n bx n aT x n bT x n 1 2 1 2 [ ] x 1 n [ ] y 1 n H  [ ] [ ] ay n by n 1 2 Linear System [ ] x 2 n [ ] H y 2 n  [ ] [ ] ax n bx n 1 2 H 5

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. [ ] x 1 n [ ] y 1 n H   [ ] [ ] y n n x n n 1 0 1 0 H 6

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.          [ ] [ ] [ ] y n T x k n k     k  From principle of superposition:        [ ] [ ] [ ] y n x k T n k   k   Property of TI:    [ ] [ ] [ ] y n x k h n k   k 7

LTI Systems (Convolution) :     [ ] [ ] [ ] y n x k h n k   k  Above equation commonly called convolution sum and represented by the notation   [ ] [ ] [ ] y n x n h n 8

Convolution properties:  Commutativity:    [ ] [ ] [ ] [ ] x n h n h n x n  Associativity:      ( [ ] [ ]) [ ] [ ] ( [ ] [ ]) h n h n h n h n h n h n 1 2 3 1 2 3  Distributivity:       [ ] ( [ ] [ ]) ( [ ] [ ]) ( [ ] [ ]) h n ax n bx n a h n x n b h n x n 1 2 1 2  Time reversal:      [ ] [ ] [ ] y n x n h n 9

…Convolution properties:  If two systems are cascaded, H1 H2  The overall impulse response of the combined system is the convolution of the individual IR:   [ ] [ ] [ ] h n h n h n 1 2  The overall IR is independent of the order: H2 H1 10

Duration of IR:  Infinite-duration impulse-response (IIR).  Finite-duration impulse-response (FIR)       [ ] [ ] [ 1 ] ... [ ] y n b x n b x n b x n q 0 1 q  In this case the IR can be read from the right-hand side of:  [ ] h n b n 11

Transforms:  Transforms are a powerful tool for simplifying the analysis of signals and of linear systems.  Interesting transforms for us:  Linearity applies:    [ ] [ ] [ ] T ax by aT x bT y  Convolution is replaced by simpler operation:   [ ] [ ] [ ] T x y T x T y 12

…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) 13

The Z Transform:  Definition Equations:  Direct Z transform     n ( ) [ ] X z x n z   n  The Region Of Convergence (ROC) plays an essential role. 14

The Z Transform (Elementary functions) :  Elementary functions and their Z-transforms:   [ ] [ ] x n n  Unit impulse:       n ( ) [ ] 1 : X z n z ROC All z   n    [ ] [ ]  Delayed unit impulse: x n n k          n k ( ) [ ] : 0 X z n k z z ROC z   n 15

The Z Transform ( … Elementary functions) :   1 , n 0   Unit Step:  [ n ] u  0, otherwise  1      n ( ) : | | 1 X z z ROC z   1 1 z  0 n  n [ ] [ ] x n a u n  Exponential:  1      n n ( ) : | | | | X z a z ROC z a   1 1 az  0 n 16

Z Transform (Cont ’ d)  Important Z Transforms Region Of Convergence (ROC) Whole Page z≠0 |z| > 1 |z| > |a| 17

The Z Transform (Elementary properties) :  Elementary properties of the Z transforms:  Linearity:       [ ] [ ] ( ) ( ) ax n by n aX z bY z   [ ] [ ] [ ] w n x n y n  Convolution: if ,Then  ( ) ( ) ( ) W z X z Y z 18

The Z Transform ( … Elementary properties) :  Shifting:       k [ ] ( ) x n k z X z  Differences:  Forward differences of a function,     [ ] [ 1 ] [ ] x n x n x n  Backward differences of a function,     [ ] [ ] [ 1 ] x n x n x n 19

The Z Transform ( … Region Of Convergence for Z transform) :          [ ] [ ] [ 1 ] [ ] x n x n n n  Since the shifting theorem      [ ] ( 1 ) ( ) Z x n z X z       1 [ ] ( 1 ) ( ) Z x n z X z 20

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. 21

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    z 0 possibly or . z  If x[n] is a right-sided sequence , the ROC extends outward from the outermost finite   z ( z ) X pole in to .  The ROC must be a connected region. 22

The Z Transform (…Region Of Convergence for Z transform) :  A two-sided sequence 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 ( z ) X  0 z . 23

The Z Transform (Application to LTI systems) :   [ ] [ ] [ ] y n x n h n  We have seen that  By the convolution property of the Z transform  ( ) ( ) ( ) Y z X z H z  Where H(z) is the transfer function of system.  Stability  | [ ] | x n M  A system is stable if a bounded input produced a bounded output, and a LTI system    | [ ] | h k is stable if: k 24

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

Discrete-time Fourier Transform       j j n ( ) [ ] X e x n e   n The same as Z-transform with z on the unit circle Continuous in Frequency, periodic with period = 2*pi 26

The Discrete Fourier Transform (DFT)  Discrete Fourier transform    2 j kn 1 N   [ ] [ ] N X k x n e   0 n 2 j  N W e  It is customary to use the N  Then the direct form is:  1 N    nk [ ] [ ] X k x n W N  0 n 27

The Discrete Fourier Transform (DFT)  With the same notation the inverse DFT is  1 N 1   nk [ ] [ ] x n X k W N N  0 k 28

The DFT (Elementary functions) :  Elementary functions and their DFT:   [ ] [ ] x n n  Unit impulse:  [ ] 1 X k    [ ] [ ]  Shifted unit impulse: x n n p   kp [ ] X k W N 29

The DFT ( … Elementary functions) :  Constant:  [ ] 1 x n   [ ] [ ] X k N k  Complex exponential:   j n [ ] x n e     N     [ ] X k N k    2 30

The DFT ( … Elementary functions) :  Cosine function:   [ ] cos 2 x n f n 0   N        [ ] [ ] [ ] X k k Nf N k Nf 0 0 2 31