Lecture 1 FIR Filtering Introduction

EE3054 Signals and Systems Lecture 1 FIR Filtering Introduction

Yao Wang Polytechnic University

Most of the slides included are extracted from lecture presentations prepared by McClellan and Schafer License Info for SPFirst Slides

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Speech waveform in time (1-D signal) Stock prices in time (1-D signal) Images (2-D signal) We will focus on 1-D signal in this class Continuous time signal x(t) Discrete time signal x[n] Discrete in nature Sampled from continuous time signal A Speech Signal:

Something that transforms an input signal to another signal Speech: Noise removal, echo cancellation, … Image: contrast enhancement, deblurring, … General mathematical description: y(t) = T ( x(t) ) y[n] = T ( x[n] ) EXAMPLES: POINTWISE OPERATORS SQUARING: y[n] = (x[n])2 RUNNING AVERAGE RULE: “the output at time n is the average of three consecutive input values” How to characterize signals?

In time domain directly In a transform domain: Represent a signal as sum of some basis signals Coefficients for different bases leads to a transform domain representation Fourier transform: Use sinusoidal signals as bases for both continuous and discrete time Z-transform: for discrete time only Laplace transform: for continuous time only How to characterize systems?

Impulse response Frequency response Helpful to look from a transformed domain representation Fourier transform: Impulse response -> frequency response Z-transform/Laplace transform: impulse response -> transfer function Topics Covered in This Course

Discrete time signals and systems Linear time invariant systems Convolution or filtering Impulse response Frequency response Z-transform analysis Discrete time Fourier transform Continuous time signals and systems Sampling of continuous time signals Using MATLAB for implementing signal and system operations and transforms LECTURE OBJECTIVES

INTRODUCE FILTERING IDEA Weighted Average Running Average FINITE IMPULSE RESPONSE FILTERS FIRFIR Filters Show how to compute the output y[n] from the input signal, x[n]

x(t)x[n] y[n] y(t) A-to-D COMPUTER D-to-A

CONCENTRATE on the COMPUTER PROCESSING ALGORITHMS SOFTWARE (MATLAB) HARDWARE: DSP chips, VLSI DSP: DIGITAL SIGNAL PROCESSING

x[n] y[n] COMPUTER

OPERATE on x[n] to get y[n] WANT a GENERAL CLASS of SYSTEMS ANALYZE the SYSTEM TOOLS: TIME-DOMAIN & FREQUENCY- DOMAIN SYNTHESIZE the SYSTEM

x[n] y[n] SYSTEM

EXAMPLES: POINTWISE OPERATORS SQUARING: y[n] = (x[n])2 RUNNING AVERAGE RULE: “the output at time n is the average of three consecutive input values”

x[n] is a LIST of NUMBERS INDEXED by “n”

STEM PLOT

n=0

= 1 + + + + y[n] 3 (x[n] x[n 1] x[n 2])

OUTPUT SIGNAL

Uses “PAST” VALUES of x[n] IMPORTANT IF “n” represents REAL TIME WHEN x[n] & y[n] ARE STREAMS

= 1 + − + − y[n] 3 (x[n] x[n 1] x[n 2])

The output signals in previous examples are shifted from the input signals To align the output with the input, use both “PAST” and “FUTURE” VALUES of x[n] = 1 − + + + y[n] 3 (x[n 1] x[n] x[n 1])

Go through on the board PAST, PRESENT, FUTURE

“n”is TIME

USE PAST VALUES

CAUSAL: Use Previous

INPUT

OUTPUT

3 = − y[n] ∑bk x[n k] k=0 = 3x[n] − x[n −1] + 2x[n − 2] + x[n − 3]

Given an input signal Compute the output of the previous filter

SLIDE a WINDOW across x[n] M = − y[n] ∑bk x[n k] k=0

Input signal from 0 to N-1, length=L1=N Filter from 0 to M, length = L2= M+1 Output signal? From 0 to N+M-1, length L3=N+M=L1+L2-1 Causal vs. Non-causal Filters?

So far we only considered the filters that range from 0 to M Output at n depends on input from n-M to n (past values only) These are called “Causal” filters More generally, the filter may range from M1 to M2 Output at n depends on input from n-M2 to n-M1 Ex. Centered averaging over 7 points, M1=-3, M2=3 Unit Impulse Signal

x[n] has only one NON-ZERO VALUE

1 n = 0 δ [n] =  0 n ≠ 0

UNIT-IMPULSE 1

n = 3 δ[n] is NON-ZERO δ [n − 3] When its argument is equal to ZERO

Use SHIFTED IMPULSES to write x[n] x[n] = 2δ [n] + 4δ [n −1] + 6δ [n − 2] + 4δ [n − 3] + 2δ [n − 4]

This formula ALWAYS works

CAUSAL SYSTEM: USE PAST VALUES y[n] = 1 (x[n] + x[n −1] + x[n − 2] + x[n − 3]) 4 δ INPUT = UNIT IMPULSEδ SIGNAL = δ[n] x[n] = δ [n] δ = 1 + 1 − + 1 − + 1 δ − y[n] 4 [n] 4 [n 1] 4 [n 2] 4 [n 3] OUTPUT is called “IMPULSE RESPONSE” = 1 1 1 1 h[n] { , 0, 0, 4 , 4 , 4 , 4 , 0, 0, }

= 1 + − + − + − y[n] 4 (x[n] x[n 1] x[n 2] x[n 3]) δ[n] “READS OUT” the FILTER COEFFICIENTS = 1 1 1 1 h[n] { , 0, 0, 4 , 4 , 4 , 4 , 0, 0, } “h” in h[n] denotes Impulse Response n=0 NON-ZERO n=–1 When window δ n=0 1 overlaps [n] n=1 n n=4 n=5 1/22/2008 © 2003, JH McClellan & RW Schafer 33 What is Impulse Response?

Impulse response is the output signal when the input is an impulse Finite Impulse Response (FIR) system: Systems for which the impulse response has finite duration For FIR system, impulse response = Filter coefficients h[k] = b_k Output = h[k]* input

Convolution = Filter Definition Filter Coeffs = Impulse Response

M M = − = − y[n] ∑bk x[n k] y[n] ∑h[k]x[n k] k=0 k=0 CONVOLUTION

Use SHIFTED IMPULSES to write h[n] h[n] = δ [n] −δ [n −1] + 2δ [n − 2] −δ [n − 3] +δ [n − 4]

2 h[n] 1

0 4 n = − − –1 bk {1, 1, 2, 1,1}

OutputOutput == ConvolutionConvolution ofof x[n]x[n] && h[n]h[n] NOTATION: y[n] = h[n]∗ x[n] Here is the FIR case: FINITE LIMITS

M y[n] = ∑h[k]x[n − k] k=0 FINITE LIMITS Same as bk

1/22/2008 © 2003, JH McClellan & RW Schafer 37 CONVOLUTION Example h[n] = δ [n] −δ [n −1] + 2δ [n − 2] −δ [n − 3] + δ [n − 4] x[n] = u[n] n −101234567 x[n]01111111... h[n]01−12−11000 h[0]x[n] 011111111 h[1]x[n −1] 00−1 −1 −1 −1 −1 −1 −1 h[2]x[n − 2] 000222222 h[3]x[n − 3] 00 0 0−1 −1 −1 −1 −1 h[4]x[n − 4] 000001111

Another example

SLIDE a Length-L WINDOW over x[n] When h[n] is not symmetric, needs to flip h(n) first!

x[n-M] x[n]

FIR Filter is “FIRST DIFFERENCE” y[n] = x[n] - x[n-1] INPUT is “UNIT STEP” 1 n ≥ 0 u[n] =  0 n < 0

Find y[n] y[n] = u[n] − u[n −1] = δ [n]

This Lecture: Chapter 1 Chapter 5, Sects. 5-1, 5-2 and 5-3 (partial)