DSP2 - EEL 5525: Foundations of DSP Topic 2 Discrete-Time...

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Unformatted text preview: EEL 5525: Foundations of DSP Topic 2 Discrete-Time Signals and Systems Dr. Liuqing Yang Department of ECE University of Florida Email: lqyang@ece.ufl.edu http://www.yang.ece.ufl.edu 1 Dr. Liuqing Yang Outline 2.1 Review of continuous-time signals and systems 2.2 Discrete-time signals 2.3 Discrete-time systems 2 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Roadmap 2.1 Review of continuous-time signals and systems – – – – Continuous-time signals Fourier transform Classification of continuous-time systems Representation of LTI systems 2.2 Discrete-time signals 2.3 Discrete-time systems 3 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Continuous-Time Signals • Dirac delta function • Sinc function • Gaussian function 4 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Fourier Transform • Fourier transform (FT) – Sufficient condition: • Inverse FT (IFT) – Sufficient conditions (Dirichlet conditions): • x(t) satisfies • x(t) is continuous, except for discontinuity points whose number on any finite interval is finite; the limits at both sides of each discontinuity point exist • x(t) has a finite number of extrema on any finite interval 5 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Fourier Transform Pairs Delta vs. DC Sinc vs. Rect Gaussian function 6 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang The Impulse Train 7 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Systems • Single-input single-output (SISO) systems • Memoryless systems • Linear SISO systems: additivity & homogeneity • Time-invariant systems: shift-invariant • Linear time-invariant (LTI) systems 8 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Representation of LTI Systems • Impulse response • Frequency response 9 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Roadmap 2.1 Review of continuous-time signals and systems 2.2 Discrete-time signals – Sequences – Operations of sequences – Special sequences 2.3 Discrete-time systems 10 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Discrete-Time Signals: Sequences 11 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Example Sequences 12 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Operations on Sequences • Product: • Sum: • Scalar multiplication: • Unit delay: 13 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang FIR Filter Example 14 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang IIR Filter Example 15 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang IIR Filter Example: Alternative Design 16 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang IIR Filter Simulation Simulation for b=a=[1 1 1] x[n] x[n-1] x[n-2] y[n] y[n-1] y[n-2] 1 2 3 4 5 6 7 x = 1:10; b = [1 1 1]; a = [1 1 1]; y = filter(b,a,x) 17 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Upsample & Downsample • Upsample by a factor of L: ↑L – Insert (L-1) zeros between every sample of the input sequence. – MATLAB: upsample(x,L) • Downsample by a factor of M: ↓M – Keep every Mth sample of the input sequence. – MATLAB: downsample(x,M) 18 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Special Sequences • Unit sample sequence • Delayed unit sample sequence 19 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Special Sequences • Unit step sequence • Delayed unit step sequence 20 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Special Sequences • Real sinusoidal sequence A: amplitude, w0: angular frequency, f: phase • Period: Integer N ≥ 1 such that w0N=2pr, where integer r ≥ 0. • Fundamental Period: The smallest N which satisfies the above condition. 21 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Random Sequence • Generating random numbers in MATLAB – rand(M,N) MxN matrix of uniformly distributed random numbers between 0 and 1 – randn(M,N) MxN matrix of Gaussian distributed random numbers with zero mean and unit variance 22 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang >> >> >> >> h = randn(8,1)+j*randn(8,1); figure(1); plot(abs(h)) grid; axis([1 8 -1 3]) xlabel('n'); ylabel('|h[n]|'); 3 2.5 2 |h[n]| 1.5 1 0.5 0 -0.5 -1 1 2 3 4 5 6 7 8 n 23 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang >> >> >> >> x1 = sign(rand(256,1)-.5) figure(2); plot(x1) axis([0 256 -1.5 1.5]); xlabel('n'); ylabel('x_1[n]'); 1.5 1 1 x [n] 0.5 0 -0.5 -1 -1.5 0 50 100 150 n 200 250 24 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Roadmap 2.1 Review of continuous-time signals and systems 2.2 Discrete-time signals 2.3 Discrete-time systems – Classification of discrete-time systems – Interconnected systems – Linear constant-coefficient difference equations 25 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Linear System • A system is linear if superposition holds: – Suppose input x1[n] gives output y1[n] – and input x2[n] gives output y2[n]. – When the input is x[n]=ax1[n]+bx2[n] – the system is said to be linear if the output is y[n]=ay1[n]+by2[n] 26 26 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Time-Invariant System • Also known as (a.k.a.) Shift-Invariance – Suppose input x[n] gives output y[n] – When the input is x[n-m] – the system is said to be time-invariant if the output is y[n-m] 27 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang LTI System A system is said to be LTI if it is both linear and time-invariant. • If input is the unit sample d[n], the output is called the impulse response h[n]. • If input is the unit step function m[n], the output is called the step response s[n]. 28 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang LTI System Representation • An arbitrary input sequence • For an LTI system, superposition holds 29 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Examples of LTI Systems • Ideal delay • Moving average • Accumulator • Forward difference • Backward difference 30 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Causal System • A system is causal if the output sample y[m] only depends on input samples x[n] for n≤m. 31 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Stable System • Bounded-input bounded-output (BIBO) stable: • For any Q: When is an LTI system stable? A: If and only if 32 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Cascade/Parallel System Connections • Cascade (a.k.a. Series) connection of systems: • Parallel connection of systems 33 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Linear Constant Coefficient Difference Equations • A class of LTI systems satisfying • usually normalized such that d0=1 • Recursive solution • Generally IIR filter, unless dk=0, k • Total solution: y[n] = yc[n]+yp[n] • yc[n]: complementary/homogeneous solution • yp[n]: particular/forcing solution 34 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Auxiliary Conditions in Linear Constant-Coefficient Difference Equations • Required to uniquely determine the output for a given input. • Can be given as N sequential values of the output. • Can be used to recursively compute later and prior values of the output. • Determines the L, TI and causality of the system. 35 © Dr. Liuqing Yang, University of Florida Dr. Liuqing Yang Summary • FT of an Impulse Train is an Impulse Train • Some special discrete-time signals – Fundamental period of discrete-time sinusoidal signals • Discrete-time systems: – I/O relationship, – Linearity, Time invariance – Causality, Stability • Linear constant coefficient difference equations – Recursive solution – Total solution – Auxiliary conditions • Some MATLAB functions 36 © Dr. Liuqing Yang, University of Florida ...
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