MM3FC Mathematical Modeling 3LECTURE 2

Times

Weeks 7,8 & 9.Lectures : Mon,Tues,Wed 10-11am,

Rm.1439Tutorials : Thurs, 10am, Rm. ULT.

Clinics : Fri, 8am, Rm.4.503Dr. Charles Unsworth,Department of Engineering Science, Rm. 4.611

Tel : 373-7599 ext. 2461Email : c.unsworth@auckland.ac.nz

This LectureWhat are we going to cover &

Why ?• Discrete representation of continuous signals. (because this is how we digitize a signal)

• The Running Average Filter.(the simplest form of FIR filter)

• The General FIR filter.

• Impulse response of a filter. (a way to classify a filter’s characteristics)

• Real world (analogue) signals are “ continuous ” in time. • A ‘continuous sinusoid’ x(t) is represented :

x(t) = Acos(ωt + φ)

• Any recorded signal is said to be “discrete ”. • A ‘discrete (digital) signal’ is a ‘snap-shot’ x[n] of a continuous

signal taken every (Ts) secs.

x[n] = x(t) with t= nTs x[n] = Acos(ω(nTs) + φ)

Where, (n) is an integer indicating the position of the values in the sequence.

Continuous & Discrete Signals

… (2.1)

Discrete-Time (Digital) Filters

• A ‘digital filter’ - takes in a digital ‘input signal’ x[n]. - alters it in some way, by an operation T{ }.- And creates a digital output signal

y[n] = T{x[n]}

x[n]

Input

y[n] = T{x[n]}

Ouput

Discrete TimeSystemT{ }

• A discrete-time (digital) signal is just a sequence of numbers.•Thus, one can compute the values of the output sequence y[n] from it’s input sequence x[n].

Example 1. y[n] = (x[n])2

The y[n] value is just the square of the x[n] value.(y[1] = x[1]2, y[2] = x[2]2, y[3] = x[3]2, …, y[n] = x[n]2).

Example 2. y[n] = max{x[n], x[n-1], x[n-2]}

The y[n] value is the largest of either x[n], x[n-1] or x[n-2].

• Obviously, there are an infinite number of systems that can be created.

Example 3 : The digital signal x[n] = {1,2,3,4,5, … , N} is passed through the filter y[n] =(x[n])2 . Determine the new filtered sequence y[n].

The ‘Running Average’ filter• A ‘Finite Impulse Response’ (FIR) Filter takes a finite length input

sequence x[n] and produces a finite length output sequence y[n].• The simplest FIR filter is the ‘running average’ filter. It computes the

‘moving average’ of two or more consecutive numbers in a sequence. • The ‘FIR’ filter is a generalisation of the idea of the running average

filter.

Example 4 : A ‘3-point running average’ filter creates 1 output value from

3 consecutive input values divided by 3.

y[0] = 1/3( x[0] + x[1] + x[2] )y[1] = 1/3( x[1] + x[2] + x[3] )

Which generalises to :

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

• This is known as a ‘difference equation’.

• It completely describes the entire ‘output signal’ for all indexed values –inf to +inf.

… (2.2)

Let’s consider a finite input signal x[n] which is triangular in shape.

Which is the discrete-time (digital) sequence ;

n n< -2 -2 -1 0 1 2 3 4 5 n>5

x[n] 0 0 0 2 4 6 4 2 0 0

• Using the difference equation : y[n] = 1/3( x[n] + x[n+1] + x[n+2] )

• We can create a ‘difference table’ to calculate the filters output.

2

4

6

4

2

N n< -2 -2 -1 0 1 2 3 4 5 n>5

x[n] 0 0 0 2 4 6 4 2 0 0

y[n] 0 2/3 2 4 14/3 4 2 2/3 0 0

2

4

6

4

2 2

4

14/3

4

2

2/3 2/3

• Note : How the output sequence is longer and smoother than the input sequence.• Note : That the output sequence starts before the input sequence starts. This is because we are using inputs from the present and future values.

• A ‘filter’ that uses ‘future’ values of the input is called ‘non-causal’.(Non-causal systems cannot be used in real-time applics. as the data is not available.)

• A ‘filter’ that uses only ‘present’ and ‘past’ values is called a ‘causal’ filter

Example 5 : Write the difference equation for a ‘3-point running average’ causal filter. And draw out it’s difference table.

Difference equation : y[n] = 1/3( x[n] + x[n-1] + x[n-2] )

N n< -2 -2 -1 0 1 2 3 4 5 6 7 n>7

x[n] 0 0 0 2 4 6 4 2 0 0 0 0

y[n] 0 0 0 2/3 2 4 14/3 4 2 2/3 0 0

• The ‘causal’ filter is also known as a ‘backward averager’ since all values are past and present.• When ‘present’ and ‘past’ values are used, in a causal filter, the output does not start before the input starts.

Causal : Uses present & past and output doesn’t start before input

Non-causal : Uses contains future and output starts before input.

• The ‘support’ of a system is the set of values where the sequence becomes non-zero.

• Thus, the ‘support’ of the causal system’sinput is :

Finite in the range :

• Thus, the ‘support’ of the causal system’s output is :

Finite in the range :

What’s the support for the ‘non-causal’ systems input and output ?

6 n 0

4 n 0

The General FIR filter• The ‘running averager’ was a special case of the ‘FIR general

difference equation’ :

• Namely, when M = 2. This is the ‘order’ of the filter.

• bk= {1/3, 1/3, 1/3}. Is the value of the ‘filter coefficient’ for each (k). (If (bk) is not the same then we say that the filter is a ‘weighted running averager’.)

• The ‘filter length’ is defined as L = M+1 = no.filter coefficients.• There will be an interval of ‘M’ samples at the beginning where the

sliding window engages with the data using less than M+1 points.• There will be an interval of ‘M’ points at the end where the sliding

window dis-engages with the data.• The ouput sequence can be as much as ‘M’ samples longer that

the input.

∑M

0=kk ]-[b = y[n] knx

Let k be –ve for a non-causal.

… (2.3)

Example 6 : The FIR filter is completely defined once the set of filter co-efficients {bk} are known.

0 1 2 3 = k{bk} = { 3, -1, 2, 1 }

• The length of the FIR filter is : L = 4.• The order of the filter is : M = L-1 = 3.

]3-[+]2-[2+]1-[-][3=

]-[b = y[n] ∑3

0=kk

nxnxnxnx

knx

Example 7: Compute the output y[n] for this FIR filter in the form of a difference table for our previous input sequence in Example 5.

• In a real-time system, we don’t have any data at time n< 0 ?• But our filter requires x[n-1], x[n-2] & x[n-3] to be known.• Intially, these values do not exist.

• The 1st ‘Initial Rest Assumption’ helps us alleviate this :

Initial Rest Assumption 1) The input x[n] is assumed to be zero prior to some starting point (n0)

i.e x[n] = 0 for n< n0

We say that the inputs are ‘suddenly’ applied.

The Unit Impulse Sequence

• The ‘unit impulse’ has the simplest sequence which consists of one nonzero value at n=0.

• The mathematical notation is the ‘kronecker delta’, [n].

n n< -2 -2 -1 0 1 2 3 4 5 6 7 n>7

[n] 0 0 0 1 0 0 0 0 0 0 0 0

0≠0

0=1=][

n

nnδ { … (2.4)

• The ‘Shifted’ Impulse response For example [n-3], is nonzero when its argument is zero.

i.e. n-3 = 0

n n< -2 -2 -1 0 1 2 3 4 5 6 7 n>7

[n-3] 0 0 0 0 0 0 1 0 0 0 0 0

• Why learn such a trivial signal ? … What is all this for … ?

Example 8 : The digital signal x[n] is respresented as the following series of unit impulses. Determine the original sequence of x[n].

x[n] = 2[n] + 4[n-1] + 6[n-2] + 4[n-3] + 2[n-4]

• Rule : By multiplying a unit impulse we multiple its magnitude.

• Compute the signal x[n] from its series of unit impulses sequence.

n n< -2 -2 -1 0 1 2 3 4 5 6 7 n>7

2[n]4[n-1] 6[n-2]4[n-3]2[n-4]x[n] =

• Thus, we can express ‘any’ digital signal as a series of ‘shifted’ unit impulses which have be scaled by a multiplication factor.

...+]1-[]1[+][]0[+]1+[]-1[+...=

]-[x = x[n] ∑k

k

nxnxnx

kn

δδδ

δ … (2.5)

The Unit Impulse Response Sequence

• When the input to an FIR filter is a unit impulse sequence, i.e. x[n] = [n], the output is by definition a ‘unit impulse response‘ sequence, i.e y[n] = h[n].

• Then the ‘general difference equation’ becomes.

∑

∑M

0=kk

M

0=kk

]-[b=][

]-[b = y[n]

knnh

knx

δ

x[n]

[n]

y[n]

h[n]

Discrete-TimeFIR filter

… (2.6)

Example 9 : Take our 3-point running averager again.y[n] = 1/3{ x[n] + x[n+1] + x[n+2]

• It’s filter coefficients are : bk = { 1/3, 1/3, 1/3 } for k = 0, 1 & 2.

2]-[ 1/3+ 1]-[ 1/3 + ][ 1/3 =

2]-[b+1]-[b + 0]-[=

]-[b=][

211

2

0=kk∑

nnn

nnnb

knnh

δδδ

δδδ

δ

Essentially, all we do is take the filter co-efficients and mutiply them by 1 at the place in the sequence they occur !

13

13

13

• Thus, by passing a unit impulse through ‘any’ filter we determine the pure response of the filter for a unit input.

Example 10 : An order 10 FIR filter is defined below. Write down the impulse response of the filter. Expand the equation and plot its impulse response.

∑10

0=k

]-[k x=][ knny

The Unit Delay • Another trivial system that

has great power is the ‘unit delay’.

• It ‘shifts’ or ‘delays’ a sytem by an amount (n0). Such that :

y[n] = x[n – n0]

• When n0 = 1, the system is called a ‘unit delay’.

• In a plot of y[n] the values of x[n] occur one time interval after they do in the input.

… (2.7)

Example 11 : A system produces a delay of 3. a) Write down the difference equation of the system. b) Calculate the output y[n] of the system in a difference table. c) Plot the input and ouput of the delay 3 system.d) Write down the impulse response equation of the system e) Plot the impulse response of the system

It has filter coefficients arew : bk = { 0, 0, 0, 1 }.

NOTE ** : Don’t be fooled that this has only 1 coefficient. It has order M=3 and has length L =M+1 = 4. But coefficients 0,1,2 are weighted to zero.

a) The difference equation is :

y[n] = 0.x[n] + 0.x[n-1] + 0.x[n-2] + 1.x[n-3] = x[n-3]

b) The differnce table is :

n n< -2 -2 -1 0 1 2 3 4 5 6 7 n>7

x[n] 0 0 0 2 4 6 4 2 0 0 0 0

y[n] 0 0 0 0 0 0 2 4 6 4 2 0

c) The input and output from the delay 3 system.

d) For the impulse response. Just replace y[n] with h[n] and x[n] with [n].

Thus, y[n] = x[n-3] h[n] = [n-3] 1 n=3 0 n3

e) The impulse response of the delay 3 system.

{