MM3FC Mathematical Modeling 3 LECTURE 2

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MM3FC Mathematical Modeling 3LECTURE 2

• Times
• Weeks 7,8 9.
• Lectures Mon,Tues,Wed 10-11am, Rm.1439
• Tutorials Thurs, 10am, Rm. ULT.
• Clinics Fri, 8am, Rm.4.503

Dr. Charles Unsworth, Department of Engineering
Science, Rm. 4.611 Tel [phone] ext.
2461 Email c.unsworth_at_auckland.ac.nz
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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 filters characteristics)

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Continuous Discrete Signals

• Real world (analogue) signals are continuous
  in time.
• A continuous sinusoid x(t) is represented
• x(t) Acos(?t f)
• Any recorded signal is said to be discrete .
  
• A discrete (digital) signal is a snap-shot
  xn of a continuous signal taken every (Ts)
  secs.
• xn x(t) with t nTs xn
  Acos(?(nTs) f)
• Where, (n) is an integer indicating the position
  of the values in the sequence.

(2.1)
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Discrete-Time (Digital) Filters

• A digital filter
• - takes in a digital input signal xn.
• - alters it in some way, by an operation T .
• - And creates a digital output signal
  yn Txn

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• A discrete-time (digital) signal is just a
  sequence of numbers.
• Thus, one can compute the values of the output
  sequence yn from its input sequence xn.
• Example 1. yn (xn)2
• The yn value is just the square of the xn
  value.
• (y1 x12, y2 x22, y3 x32, ,
  yn xn2).
• Example 2. yn maxxn, xn-1, xn-2
• The yn value is the largest of either xn,
  xn-1 or xn-2.
• Obviously, there are an infinite number of
  systems that can be created.

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Example 3 The digital signal xn 1,2,3,4,5,
, N is passed through the filter yn (xn)2
. Determine the new filtered sequence yn.
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The Running Average filter

• A Finite Impulse Response (FIR) Filter takes a
  finite length input sequence xn and produces a
  finite length output sequence yn.
• 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.
• y0 1/3( x0 x1 x2 )
• y1 1/3( x1 x2 x3 )
• Which generalises to
• yn 1/3( xn xn1 xn2 )
• This is known as a difference equation.
• It completely describes the entire output
  signal for all indexed values inf to inf.

(2.2)
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• Lets consider a finite input signal xn which
  is triangular in shape.
• Which is the discrete-time (digital) sequence
• n nlt -2 -2 -1 0 1 2 3 4 5 ngt5
• xn 0 0 0 2 4 6 4 2 0 0
• Using the difference equation yn 1/3( xn
  xn1 xn2 )
• We can create a difference table to calculate
  the filters output.

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N nlt -2 -2 -1 0 1 2 3 4 5 ngt5 xn 0 0
0 2 4 6 4 2 0 0 yn 0 2/3 2 4
14/3 4 2 2/3 0 0

• 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.

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• 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 its difference table.
• Difference equation yn 1/3( xn xn-1
  xn-2 )
• N nlt -2 -2 -1 0 1 2 3 4 5 6
  7 ngt7
• xn 0 0 0 2 4 6 4 2 0 0 0
  0
• yn 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.

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Causal Uses present past and output doesnt
start before input
Non-causal Uses contains future and output
starts before input.
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• The support of a system is the set of values
  where the sequence becomes non-zero.

• Thus, the support of the causal systems
• input is
• Finite in the range
• Thus, the support of the causal systems
  output is
• Finite in the range
• Whats the support for the non-causal systems
  input and output ?

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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 M1
  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 M1 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.

(2.3)
Let k be ve for a non-causal.
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• 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.

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Example 7 Compute the output yn for this FIR
filter in the form of a difference table for our
previous input sequence in Example 5.
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• In a real-time system, we dont have any data at
  time nlt 0 ?
• But our filter requires xn-1, xn-2 xn-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 xn is assumed to be zero prior to
  some starting point (n0)
• i.e xn 0 for nlt n0
• We say that the inputs are suddenly
  applied.

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The Unit Impulse Sequence

• The unit impulse has the simplest sequence
  which consists of one nonzero value at n0.
• The mathematical notation is the kronecker
  delta, ?n.
• n nlt -2 -2 -1 0 1 2 3 4
  5 6 7 ngt7
• ?n 0 0 0 1 0 0 0 0 0 0 0
  0

(2.4)
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• The Shifted Impulse response
• For example ?n-3, is nonzero when its
  argument is zero.
• i.e. n-3 0

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

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• Example 8 The digital signal xn is
  respresented as the following series of unit
  impulses. Determine the original sequence of
  xn.
• xn 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 xn from its series of unit
  impulses sequence.
• n nlt -2 -2 -1 0 1 2 3 4 5
  6 7 ngt7
• 2?n
• 4?n-1
• 6?n-2
• 4?n-3
• 2?n-4
• xn
• Thus, we can express any digital signal as a
  series of shifted unit impulses which have be
  scaled by a multiplication factor.

(2.5)
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The Unit Impulse Response Sequence

• When the input to an FIR filter is a unit impulse
  sequence, i.e. xn ?n, the output is by
  definition a unit impulse response sequence,
  i.e yn hn.
• Then the general difference equation becomes.

(2.6)
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• Example 9 Take our 3-point running averager
  again.
• yn 1/3 xn xn1 xn2
• Its filter coefficients are bk 1/3,
  1/3, 1/3 for k 0, 1 2.

1 3
1 3
1 3
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• 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.

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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
• yn xn n0
• When n0 1, the system is called a unit delay.
  
• In a plot of yn the values of xn occur one
  time interval after they do in the input.

(2.7)
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• Example 11 A system produces a delay of 3.
• a) Write down the difference equation of the
  system.
• b) Calculate the output yn 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 Dont be fooled that this has only 1
  coefficient. It has order M3 and has length L
  M1 4. But coefficients 0,1,2 are weighted to
  zero.
• a) The difference equation is
• yn 0.xn 0.xn-1 0.xn-2 1.xn-3
• xn-3
• b) The differnce table is
• n nlt -2 -2 -1 0 1 2 3 4 5 6
  7 ngt7
• xn 0 0 0 2 4 6 4 2 0 0 0
  0

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c) The input and output from the delay 3
system. d) For the impulse response.
Just replace yn with hn and xn with
?n. Thus, yn xn-3 ?
hn ?n-3 1 n3 0
n?3 e) The impulse response of the delay 3
system.