LTI_systems_w10

LTI_systems_w10 - S Y S C 4 4 0 5 : D i g i t a l S i g n a...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
SYSC4405: Digital Signal Processing LTI_systems_w10.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 1 / 54 Linear Time Invariant Systems 1 Discrete Time Systems A discrete-time system is a device or an algorithm that operates on an input sequences x(n) to produce an output sequence y(n) according to some rule or a computational procedure. PROPERTIES 1. Linearity A discrete-time system is linear if the following holds true: 1. Sections 2.3, 2.4 and 2.5 in Proakis and Manolakis discrete-time System Input sequence x(n) Output sequence y(n) If an input sequence x 1 (n) produces an output sequence y 1 (n) and If an input sequence x 2 (n) produces an output sequence y 2 (n) then α 1 x 1 (n)+ α 2 x 2 (n) produces an output α 1 y 1 (n)+ α 2 y 2 (n) where α 1 and α 2 are arbitrary constants. In other words, the input/output relationship obeys the principles of scaling and additivity.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
SYSC4405: Digital Signal Processing LTI_systems_w10.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 2 / 54 Testing if "System" is linear: 2. Time Invariance A discrete-time system is considered time-invariant if its response does not depend on an absolute time reference System x 1 n () y 1 n X α 1 + System x 2 n y 2 n X α 2 wn x 1 n X α 1 + System x 2 n X α 2 yn The system is linear if =
Background image of page 2
SYSC4405: Digital Signal Processing LTI_systems_w10.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 3 / 54 .In other words: If an input sequence x 1 (n) produces an output sequence y 1 (n) and If an input sequence x 2 (n) produces an output sequence y 2 (n) and let x 2 (n) = x 1 (n-n 0 ) If the system is time invariant, then y 2 (n) = y 1 (n-n 0 ) n n n 0 x(n) y(n) x(n-n 0 ) y(n-n 0 ) n n n 0
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
SYSC4405: Digital Signal Processing LTI_systems_w10.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 4 / 54 Testing if "System" is time-invariant: "System" is time-invariant only if: 0.1 3. Linear-Time-Invariance A discrete-time system is called a linear-time-invariant (LTI) system if it is both linear and time-invariant. 4. Causality A causal system is non-predictive. In other words, the output sample at time n=N depends only on {. ...... x(N-2), x(N-1), x(N)}; i.e. on the input samples x(n) for . System under test Delay by n 0 Delay by n 0 System under test xt () xt t 0 yt wn yn n 0 0 = nN
Background image of page 4
SYSC4405: Digital Signal Processing LTI_systems_w10.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 5 / 54 5. Stability 1 To show that a system is unstable, it suffices to show that there is at least one bounded input sequence for which the output becomes unbounded. Example: A discrete-time system is described by the input/output relationship: for Is this system linear? time invariant? stable? causal?
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 04/14/2010.

Page1 / 54

LTI_systems_w10 - S Y S C 4 4 0 5 : D i g i t a l S i g n a...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online