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EE3054 Signals and Systems

Lecture 1FIR Filtering Introduction

Yao WangPolytechnic University

Most of the slides included are extracted from lecture presentations prepared by McClellan and Schafer

1/22/2008 © 2003, JH McClellan & RW Schafer 2

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What is a signal?

� 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:

What is a system?

� 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]


DIGITAL FILTERING

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

� DSP: DIGITAL SIGNAL PROCESSING

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


DISCRETE-TIME SYSTEM

� 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

COMPUTERy[n]x[n]


D-T SYSTEM EXAMPLES

� 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”

SYSTEMy[n]x[n]


DISCRETE-TIME SIGNAL

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

STEM PLOT


3-PT AVERAGE SYSTEM� ADD 3 CONSECUTIVE NUMBERS

� Do this for each “n”Make a TABLE

n=0

n=1

])2[]1[][(][ 31 ++++= nxnxnxny


INPUT SIGNAL

OUTPUT SIGNAL

])2[]1[][(][ 31 ++++= nxnxnxny


ANOTHER 3-pt AVERAGER

� Uses “PAST” VALUES of x[n]� IMPORTANT IF “n” represents REAL TIME

� WHEN x[n] & y[n] ARE STREAMS

])2[]1[][(][ 31 −+−+= nxnxnxny

ANOTHER 3-pt AVERAGER

� 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]

� Go through on the board

])1[][]1[(][ 31 +++−= nxnxnxny


PAST, PRESENT, FUTURE

“n” is TIME


3-pt AVG EXAMPLE

USE PAST VALUES

400for)4/8/2cos()02.1(][ :Input ≤≤++= nnnx n ππ


7-pt AVG EXAMPLE

CAUSAL: Use Previous

LONGER OUTPUT

400for)4/8/2cos()02.1(][ :Input ≤≤++= nnnx n ππ


FILTERED STOCK SIGNAL

OUTPUT

INPUT

50-pt Averager


GENERAL FIR FILTER

� FILTER COEFFICIENTS {bk}� DEFINE THE FILTER

� For example,

∑=

−=M

kk knxbny

0][][

]3[]2[2]1[][3

][][3

0

−+−+−−=

−=∑=

nxnxnxnx

knxbnyk

k

}1,2,1,3{ −=kb


Example

� Given an input signal� Compute the output of the previous filter


GENERAL FIR FILTER

� FILTER COEFFICIENTS {bk}

� FILTER ORDER is M� FILTER LENGTH is L = M+1

� NUMBER of FILTER COEFFS is L

∑=

−=M

kk knxbny

0][][


GENERAL FIR FILTER

� SLIDE a WINDOW across x[n]

x[n]x[n-M]

∑=

−=M

kk knxbny

0][][

More on signal ranges and lengths after filtering

� 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


≠

==

00

01][

n

nnδ

Unit Impulse Signal

� x[n] has only one NON-ZERO VALUE

1

n

UNIT-IMPULSE


UNIT IMPULSE SIGNAL δδδδ[n]

δδδδ[n] is NON-ZEROWhen its argumentis equal to ZERO

]3[ −nδ 3=n


]4[2]3[4]2[6]1[4][2][ −+−+−+−+= nnnnnnx δδδδδ

Any x[n] Can be Represented by Impuse Signals!

� Use SHIFTED IMPULSES to write x[n]


SUM of SHIFTED IMPULSES

This formula ALWAYS works


4-pt AVERAGER

� CAUSAL SYSTEM: USE PAST VALUES])3[]2[]1[][(][ 4

1 −+−+−+= nxnxnxnxny

� INPUT = UNIT IMPULSE SIGNAL = δδδδ[n]

]3[]2[]1[][][][][

41

41

41

41 −+−+−+=

=nnnnny

nnxδδδδ

δ

� OUTPUT is called “IMPULSE RESPONSE”},0,0,,,,,0,0,{][ 4

141

41

41

��=nh


},0,0,,,,,0,0,{][ 41

41

41

41

��=nh

4-pt Avg Impulse Response

δδδδ[n] “READS OUT” the FILTER COEFFICIENTS

n=0“h” in h[n] denotes Impulse Response

1

n

n=0n=1

n=4n=5

n=–1

NON-ZERO When window overlaps δδδδ[n]

])3[]2[]1[][(][ 41 −+−+−+= nxnxnxnxny


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


FIR IMPULSE RESPONSE

� Convolution = Filter Definition� Filter Coeffs = Impulse Response

∑=

−=M

kknxkhny

0][][][

CONVOLUTION

∑=

−=M

kk knxbny

0][][


]4[]3[]2[2]1[][][ −+−−−+−−= nnnnnnh δδδδδ

MATH FORMULA for h[n]

� Use SHIFTED IMPULSES to write h[n]

1

n

–1

2

0 4

][nh

}1,1,2,1,1{ −−=kb


∑=

−=M

kknxkhny

0][][][

Convolution Sum

�� Output = Convolution of x[n] & h[n]Output = Convolution of x[n] & h[n]� NOTATION:� Here is the FIR case:

FINITE LIMITS

FINITE LIMITSSame as bk

][][][ nxnhny ∗=


CONVOLUTION Example

n −1 0 1 2 3 4 5 6 7x[n] 0 1 1 1 1 1 1 1 ...h[n] 0 1 −1 2 −1 1 0 0 0

0 1 1 1 1 1 1 1 10 0 −1 −1 −1 −1 −1 −1 −10 0 0 2 2 2 2 2 20 0 0 0 −1 −1 −1 −1 −10 0 0 0 0 1 1 1 1

y[n] 0 1 0 2 1 2 2 2 ...

][][]4[]3[]2[2]1[][][

nunxnnnnnnh

=−+−−−+−−= δδδδδ

]4[]4[]3[]3[]2[]2[

]1[]1[][]0[

−−−−

nxhnxhnxhnxhnxh


Convolution via Synthetic Polynomial Multiplication

� Another example


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

x[n]x[n-M]


DCONVDEMO: MATLAB GUI


� Go through the demo program for different types of signals


� Length of filtered signals� Alignment of input and output


]1[][][ −−= nununy

POP QUIZ

� FIR Filter is “FIRST DIFFERENCE”� y[n] = x[n] - x[n-1]

� INPUT is “UNIT STEP”

� Find y[n]

<

≥=

00

01][

n

nnu

][]1[][][ nnununy δ=−−=

READING ASSIGNMENTS

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

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