ee338_summary_basics - EE 338 Discrete‐time Signals and...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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). • ...
View Full Document

Ask a homework question - tutors are online