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Unformatted text preview: EE 338 Discrete‐time Signals and Systems Part I: Basics of Signals and Systems, and Time‐domain Analysis 1. Basic discrete‐time signals: • • • • Unit impulse sequence , unit step sequence u[n], exponential sequence , Sinusoidal sequence sin ωn o x[n] is periodic in the time domain if and only if there exists two integers k and N such that 2 . If N and k do not have common factors, then N is the period of x[n]. o sin sin 2 , and a discrete‐time system cannot differentiate between and 2 (k is an integer). 2. Discrete‐time systems: • Basic operations on discrete‐time signals: o Folding x[‐n], shifting x[n‐m], addition x[n]+y[n], and multiplication x[n]y[n] o To find x[‐n+m] from x[n]: fold x[n] first, and then shift x[‐n] to the right (left) by |m| if m>0 (m<0) Determine the linearity, time invariance, causality, and BIBO stability of discrete‐time systems • 3. Linear time‐invariant systems: • • Unit impulse response h[n] is the system’s output given a unit impulse input sequence. h[n] uniquely characterizes the discrete‐time system. ∑ ∑ (Linear) convolution sum: o Computation of convolution sum o If x[n] is of finite duration N and h[n] is of finite duration M, then y[n] is finite duration N+M‐1. o Properties of convolution sum: Commutative law: Associative law: Distributive law: Identity property: Shifting property: Causal LTI system: an LTI system is causal if and only if h[n]=0 for all n<0 | | ∞ BIBO stability of LTI system: an LTI system is BIBO stable if and only if ∑ FIR (h[n] is of finite duration) and IIR (h[n] has infinite number of samples with non‐zero values) • • • 4. Causal constant‐coefficient difference equations: ∑ • • ∑ with 1 0, then it is called a non‐recursive system (without feedback from If previous outputs) Given a causal input sequence x[n] and the initial condition 1, 2 ,…, , the output of the system is , where is the homogeneous (zero‐input) solution is the particular (zero‐state) solution. and o The zero‐input solution 0 satisfies ∑ ∑ o The zero‐state solution satisfies ∑ with 1 2 0. The unit impulse response of an LTI recursive system is the zero‐state response of the system when the input is and when the system is relaxed ( 1 2 0). • ...
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- Spring '10