The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Review Topics February 28, 2012

Discrete-Time n Z, ω [ π, π) ∈Convolution:∈ − y = h x: ∗ ∞ y [n] = h [k] x [n k] − k= X−∞ If h has length M and x has length N then y = h x has length N + M 1 z-transform: ∗ − ∞ n X (z) = x [n] z− n= X−∞ DTFT (Discrete-Time Fourier Transform) and IDTFT:

∞ jωn X (ω) = x [n] e− n= X−∞ 1 π x [n] = X (ω) ejωndω 2π π Z− Other results and properties:

Definition of δ [n] and u [n] • Convolution theorem: y = h x has z-transform Y (z) = H (z) X (z) and DTFT • Y (ω) = H (ω) X (ω) ∗

z n z z-transform pairs: δ [n] 1, u [n] z 1 , α u [n] z α • ←→ ←→ − ←→ − z-transform and DTFT related via z = ejω • BIBO stability iff all poles inside unit circle (technically, this presumes causal systems • exclusively)

Filters:

FIR filters: definition; transversal filter structure and difference equation corresponding • 1 N to H (z) = h0 + h1z− + + hN z− ; N=order and N + 1= length ··· IIR filters: definition; direct form II and direct form II transposed structures and • difference equation (time-domain expression, not the circuit diagram) corresponding to rational function H (z); N= order= # delays in the direct form II structure

1 For case of discrete periodic time: 0 n N 1, and frequency indices 0 k N 1, apply DFT (Discrete Fourier Transform)≤ and≤ IDTFT:− ≤ ≤ −

N 1 − 2π X [k] = x [n] exp j nk − N n=0   XN 1 1 − 2π x [n] = X [k] exp j nk N N k=0 X  

Basic result of sampling theory: f= analog frequency (Hz); fs= sampling rate (Hz); ω= normalized digital radian frequency (rad):

ω = 2πf/fs

Continuous-Time t R, ω R ∈ Convolution:∈ y = h x: ∗ ∞ y (t) = h (τ) x (t τ) dτ − Z−∞ If h has duration Th and x has duration Tx then y has duration Th + Tx Laplace transform: ∞ st X (s) = x (t) e− dt Z−∞ CTFT (Continuous-Time Fourier Transform) and ICTFT:

∞ jωt X (ω) = x (t) e− dt Z−∞ 1 ∞ x (t) = X (ω) ejωtdω 2π Z−∞ Other results and properties:

Definition of δ (t) and u (t) • Convolution theorem: y = h x has z-transform Y (s) = H (s) X (s) and CTFT • Y (ω) = H (ω) X (ω) ∗

1 αt 1 Laplace transform pairs: δ (t) 1, u (t) s , e u (t) s α • ←→ ←→ ←→ − Laplace transform and CTFT related via s = jω • BIBO stability iff all poles inside LHP (technically, this presumes causal systems • exclusively)

2 LTI Systems

Definition of linear, time-invariant, causal, BIBO stability. LTI system characterized in time-domain by convolution with impulse response; in fre- quency domain, characterized by multiplication with frequency response; in transform do- main, characterized by multiplication with transfer function.

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