Xử lý tín hiệu số signal and system in time domain

Structure for the realization of LTI systems • Consider the 1st order system • This realization uses separate delays for both input and output, called Direct Form 1 structure

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Xử lý tín hiệu số Signal and System in Time Domain Ngô Quốc Cường Ngô Quốc Cường ngoquoccuong175@gmail.com sites.google.com/a/hcmute.edu.vn/ngoquoccuong Signal and System in Time Domain • Discrete time signals • Discrete time systems • LTI systems 2 2.1 Discrete - time signals • A discrete time signal x(n) is a function of an independent variable that is integer. 3 2.1 Discrete - time signals • Alternative representation of discrete time signal: 4 2.1 Discrete - time signals • Alternative representation of discrete time signal: 5 2.1.1 Some elementary signals 6 2.1.1 Some elementary signals 7 2.1.1 Some elementary signals 8 2.1.1 Some elementary signals 9 2.1.2 Classification of discrete time signal • Energy signals and power signal – The energy E of a signal x(n) is given: – If E is finite, x(n) is call an energy signal. 10 2.1.2 Classification of discrete time signal • Energy signals and power signal – The average power P of a signal x(n) is defined: – If P is finite (and nonzero), x(n) is called a power signal. 11 2.1.2 Classification of discrete time signal • Energy signals and power signal – Example: the average power of the unit step signal is: 12 2.1.2 Classification of discrete time signal • Periodic signals and aperiodic signals – A signal x(n) is periodic with period N (N >0) if and only if 13 2.1.2 Classification of discrete time signal • Symmetric (even) and antisymmetric (odd) signals – A real value signal x(n) is call symmetric if – A signal is call antisymmetric if 14 2.1.2 Classification of discrete time signal 15 • Symmetric (even) and antisymmetric (odd) signals 2.1.2 Classification of discrete time signal – The even signal component is formed by adding x(n) to x(- n) and dividing by 2. – Odd signal component 16 2.1.3 Simple manipulations of signals 17 • Transformation of time – A signal x(n) may be shifted by replacing n bay n-k. • k is positive number: delay • k is negative number: advance – Folding: replace n by -n – Time scaling: replace n by cn (c is an integer) 2.1.3 Simple manipulations of signals • Transformation of time – Find x(n-3) and x(n+2) of x(n) 18 2.1.3 Simple manipulations of signals • Transformation of time – Find x(-n) and x(-n+2) of x(n) 19 2.1.3 Simple manipulations of signals • Transformation of time – Show the graphical representation of y(n) = x(2n), where x(n) is 20 2.1.3 Simple manipulations of signals • Addition, multiplication, and scaling – Amplitude scaling – Sum – Product 21 Exercises 22 Exercises • x(n) is illustrated in the figure • Sketch the following signals 23 2.2 Discrete time systems • Device or algorithm that performed some prescribed operation on discrete time signal. 24 2.2 Discrete time systems • Determine the response of the following systems to the input signal 25 2.2 Discrete time systems • Block diagram representation – An adder 26 2.2 Discrete time systems • Block diagram representation – A constant multiplier 27 2.2 Discrete time systems • Block diagram representation – A signal multiplier 28 2.2 Discrete time systems • Block diagram representation – A unit delay element – A unit advance element 29 2.2 Discrete time systems • Block diagram representation 30 2.2 Discrete time systems • Classification of discrete time systems – Static versus dynamic systems – Time invariant versus time variant systems – Linear versus nonlinear systems – Causal versus noncausal systems – Stable versus unstable systems 31 2.2 Discrete time systems • Classification of discrete time systems – Static versus dynamic systems • Static: output at any instant n depends at most on the input sample at the same time – memoryless. • Dynamic: to have memory 32 2.2 Discrete time systems • Classification of discrete time systems – Time invariant versus time variant systems • Time invariant: input – output characteristics do not change with time. 33 2.2 Discrete time systems • Classification of discrete time systems – Linear versus nonlinear systems 34 2.2 Discrete time systems 35 2.2 Discrete time systems • Classification of discrete time systems – Causal versus noncausal systems • The output of the system at any time n depends only on present and past inputs but does not depend on future inputs. 36 2.2 Discrete time systems 37 2.2 Discrete time systems • Classification of discrete time systems – Stable versus unstable systems • An arbitrary relaxed system is said to be bounded input bounded output stable if and only if every bounded input produces a bounded output. 𝑥 𝑛 ≤ 𝑀𝑥 < ∞ 𝑦 𝑛 ≤ 𝑀𝑦 < ∞ 38 2.2 Discrete time systems 39 2.2 Discrete time systems • Interconnection of discrete time systems 40 2.3 Analysis of discrete time LTI systems • LTI: Linear Time Invariant • 2 methods: – Solve the difference equation – Decompose the input signal into a sum of elementary signals. Using the linearity property, the responses of the system to the elementary signals are added to obtain the total response of the system. 41 2.3 Analysis of discrete time LTI systems • Resolution of discrete time signal into impulses • Example: 42 2.3 Analysis of discrete time LTI systems • Response of LTI system to arbitrary input – Denote the response y(n, k) of the system to unit sample sequence at n = k by symbol h(n, k). – The response of the system to x(n) 43 2.3 Analysis of discrete time LTI systems • Response of LTI system to arbitrary input: convolution – The formula reduces to – The response at n = n0 is given as 44 2.3 Analysis of discrete time LTI systems • Response of LTI system to arbitrary input – Summarize 45 2.3 Analysis of discrete time LTI systems • Response of LTI system to arbitrary input – Example – The output is 46 Exercise 47 Exercise 48 Exercise 49 2.3 Analysis of discrete time LTI systems • Properties of convolution and interconnection 50 2.3 Analysis of discrete time LTI systems • Causal linear time invariant system • An LTI system is causal if and only if its impulse response is zero for n<0. 51 2.3 Analysis of discrete time LTI systems • Stability of linear time invariant system – A linear time invariant system is stable if its impulse response is summable. 52 • LTI system can be characterized in terms of its impulse response h(n). – Finite duration impulse response – Infinite duration impulse response • Causal FIR system 53 • Practical DSP methods fall in two basic classes: – Block processing methods. – Sample processing methods. • In block processing methods, the data are collected and processed in blocks. • In sample processing methods, the data are processed one at a time. Sample processing methods are used primarily in real-time applications. 54 Block processing • Block processing methods: – Direct form – Convolution table – LTI form – Matrix form – Flip-and-slide form – Overlap-add block convolution form 55 Block processing • In many practical applications, we sample our analog input signal (in accordance with the sampling theorem requirements) and collect a finite set of samples, say L samples, representing a finite time record of the input signal. The duration of the data record in seconds will be: 56 Block processing • The direct and LTI forms of convolution given by 57 Block processing • Direct Form – Consider a causal FIR filter of order M with impulse response h(n), n = 0, 1, . . . , M. It may be represented as a block: – For the direct form 58 Block processing • Direct Form – Consider the case of an order-3 filter and a length-5 input signal. 59 Block processing • Direct Form 60 Block processing • Direct Form 61 Block processing • Convolution table 62 Block processing • Convolution table – Folding the table, 63 Block processing • LTI form – The input signal – X can be written by 64 Block processing • LTI form – The effect of the filter is to replace each delayed impulse by the corresponding delayed impulse response 65 Block processing • LTI form 66 Block processing • Matrix form – The convolutional equations can also be written in the linear matrix form – The filter matrix H must be rectangular with dimensions 67 Block processing • Matrix form – There is also an alternative matrix form written as follows: – the data matrix X has dimension: 68 Block processing • Flip and Slide form 69 Block processing • Overlap-Add block 70 Block processing • Overlap-Add block – The input is divided into the following three contiguous blocks – Convolving each block separately with h = [1, 2, −1, 1] 71 Block processing • Overlap-Add block – aligning the output blocks according to their absolute timings and adding them up 72 Problems • Compute the convolution, y = h ∗ x, of the filter and input • Using the following three methods: – (a) The convolution table. – (b) The LTI form of convolution, arranging the computations in a table form. – (c) The overlap-add method of block convolution with length-3 input blocks. Repeat using length-5 input blocks. 73 74 75 76 2.4 Discrete time systems described by difference equations • The practical implementation of the IIR system is impossible since it requires an infinite number of memory locations, multiplications, and additions. • Practical and computationally efficient means: difference equations 77 2.4 Discrete time systems described by difference equations • Recursive and non-recursive system – Compute the cumulative average of a signal x(n) defined in the interval 0 ≤ k ≤ n – In the different way – Hence 78 Recursive system 2.4 Discrete time systems described by difference equations • Recursive and non-recursive system – Non-recursive system: depends only on the present and the past inputs. 79 2.4 Discrete time systems described by difference equations • Recursive and non-recursive system 80 2.4 Discrete time systems described by difference equations • The general form: • N: the order of the difference equation = the order of the system. 81 2.4 Discrete time systems described by difference equations • Solution of linear constant coefficient difference equation – Direct method – Indirect method (z - transform) • The direct solution method assumes that the total solution is the sum of two parts: – yh(n): homogeneous solution – yp(n): particular solution 82 2.5 Structure for the realization of LTI systems • Consider the 1st order system • This realization uses separate delays for both input and output, called Direct Form 1 structure. 83 2.5 Structure for the realization of LTI systems • Interchange the order of the recursive and non-recursive systems. 84 2.5 Structure for the realization of LTI systems • Two delay elements can be merged into one delay 85 Direct Form 2 structure 2.5 Structure for the realization of LTI systems • Direct Form 1 – M+N delays – M+N+1 multipliers 86 2.5 Structure for the realization of LTI systems • Direct Form 2 87 2.5 Structure for the realization of LTI systems 88 89 90

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