Finite Impulse Response Filters

1
Finite Impulse Response Filters
2
Discrete-Time Impulse Signal

• Let dk be a discrete-time impulse function,
  a.k.a. Kronecker delta function
• Impulse response is response of discrete-time LTI
  system to discrete impulse function
• Example delay by one sample
• Finite impulse response filter
• Non-zero extent of impulse response is finite
• Can be in continuous time or discrete time
• Also called tapped delay line (see slides 3-13,
  3-19, 5-3)

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Discrete-time Tapped Delay Line

• Impulse response hk of finite extent k 0,,
  M-1
• Block diagram (finite impulse response filter)

Discrete-time convolution
Applications of continuous-time tapped delay
lines?
4
Discrete-time Convolution Derivation

• Output yk for input xk
• Any signal can be decomposedinto sum of discrete
  impulses
• Apply linear properties
• Apply shift-invariance
• Apply change of variables

yk h0 xk h1 xk-1 ( xk
xk-1 ) / 2
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Comparison to Continuous Time

• Continuous-time convolution of x(t) and h(t)
• For each t, compute different (possibly) infinite
  integral
• In discrete-time, replace integral with summation
• For each k, compute different (possibly) infinite
  summation
• LTI system
• From impulse response and input, one can
  determine output
• Impulse response uniquely characterizes LTI system

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Convolution Demos

• Johns Hopkins University Demonstrations
• http//www.jhu.edu/signals (http//www.jhu.edu/s
  ignals)
• Convolution applet to animate convolution of
  simple signals and hand-sketched signals
• Convolving two rectangular pulses of same width
  gives triangle with width of twice the width of
  rectangular pulses(also see Appendix E for
  intermediate calculations)

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Linear Time-Invariant Systems

• Complex exponentials zk havea special property
  when theyare input into LTI systems
• Output will be same complexexponential weighted
  by H(z)
• When we specialize the z-domain to frequency
  domain, magnitude of H(z) will control which
  frequencies are attenuated or passed
• H(z) is also known as the transfer function

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Linear Time-Invariant Systems

• The Fundamental Theorem of Linear Systems
• If a complex sinusoid were input into an LTI
  system, then the output would be a complex
  sinusoid of the same frequency that has been
  scaled by the frequency response of the LTI
  system at that frequency
• Scaling may attenuate the signal and shift it in
  phase
• Example in continuous time see handout F
• Example in discrete time. Let xk e j W k,
• H(W) is discrete-time Fourier transform of
  hkH(W) is also called the frequency response

H(?)
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Frequency Response

• For continuous-time systems, response to complex
  sinusoid is
• For discrete-time systems, z-k (r e j w)k
  r-k e j w k and the response is
• For discrete-time systems, response to complex
  sinusoid is

frequency response
frequency response
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Example Ideal Delay

• Continuous Time
• Delay by T seconds
• Impulse response
• Frequency response

• Discrete Time
• Delay by 1 sample
• Impulse response
• Frequency response

yk
xk
w W T
Allpass Filter
Linear Phase
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Frequency Response

• System response to complex sinusoid e j W t for
  all possible frequencies W where W 2 p f
• Above passes low frequencies, a.k.a. lowpass
  filter
• FIR filters are only realizable LTI filters that
  can have linear phase over all frequencies
• Not all FIR filters exhibit linear phase

?H(W)
H(W)
Linearphase
stopband
stopband
W
W
Wp
Ws
-Ws
-Wp
passband
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Linear Time-Invariant Systems

• Any linear time-invariant system (LTI) system,
  whether continuous-time or discrete-time, can be
  uniquely characterized by its
• Impulse response response of system to an
  impulse OR
• Frequency response response of system to a
  complex sinusoid (e j W t or e j w k) for all
  possible frequencies OR
• Transfer function general transform of impulse
  response (Laplace transform for continuous-time
  systems and z-transform for discrete-time
  systems)
• Given one, we can find other two if they exist
• Give an impulse response that has a Laplace
  transform but not a Fourier transform? What
  about the other way?

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Mandrill Demo (DSP First)

• Five-tap discrete-time averaging FIR filter with
  input xk and output yk
• Lowpass filter (smooth/blur input signal)
• Impulse response is 1/5, 1/5, 1/5, 1/5, 1/5
• First-order difference FIR filter
• Highpass filter (sharpensinput signal)
• Impulse response is 1, -1

hk
First-order difference impulse response
k
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Mandrill Demo (DSP First)

• DSP First demos http//users.ece.gatech.edu/dspf
  irst
• From lowpass filter to highpass filter
• original image ? blurred image ?
  sharpened/blurred image
• From highpass to lowpass filter
• original image ? sharpened image ?
  blurred/sharpened image
• Frequencies that are zeroed out can never be
  recovered (e.g. DC is zeroed out by highpass
  filter)
• Order of two LTI systems in cascade can be
  switched under the assumption that computations
  are performed in exact precision

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Mandrill Demo (DSP First)

• Precision
• Input is represented as eight-bit numbers 0,255
  per image pixel (i.e. fewer than three decimal
  digits of accuracy)
• Filter coeffients represented by one decimal
  digit each
• Intermediate computations (filtering) in
  double-precision floating-point arithmetic (15-16
  decimal digits of accuracy)
• Output is represented as eight-bit number -128,
  127(i.e. fewer than three decimal digits)
• No output precision was harmed in the making of
  this demo ?

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Finite Impulse Response Filters

• Duration of impulse response hk is finite, i.e.
  zero-valued for k outside interval 0, M-1
• Output depends on current input and previous M-1
  inputs
• Summation to compute yk reduces to a vector dot
  product between M input samples in the vector
• and M values of the impulse response in
  vector
• What instruction set architecture features would
  you add to accelerate FIR filtering?

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Symmetric FIR Filters

• Impulse response often symmetric about midpoint
• Phase of frequency response is linear (slides 5-9
  to 5-11)
• Example three-tap FIR filter (M 3) with h0
  h2
• Implementation savings
• Reduce number of multiplications from M to M/2
  for even-length and to (M1)/2 for odd-length
  impulse responses
• Reduce storage of impulse response by same amount
• TI TMS320C54 DSP has an accelerator instructor
  FIRS to compute h0 ( xk xk-2 ) in one
  instruction cycle
• On most DSPs, no accelerator instruction is
  available

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Filter Design

• Specify a desired piecewise constant magnitude
  response
• Lowpass filter example
• w ? 0, wp, mag ? 1-dp, 1
• w ? ws, p, mag ? 0, ds
• Transition band unspecified
• Symmetric FIR filter design methods
• Windowing
• Least squares
• Remez (Parks-McClellan)

Lowpass Filter Example
Desired Magnitude Response
forbidden
1
1-dp
forbidden
Achtung!
forbidden
ds
w
wp
ws
p
Passband
Stopband
Transition band
Red region is forbidden
dp passband ripple ds stopband ripple
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Importance of Linear phase

• Speech signals
• Use phase differences to locate a speaker
• Once locked onto a speaker, our ears are
  relatively insensitive to phase distortion in
  speech from that speaker (underlies speech
  compression systems, e.g. digital cell phones)

• Linear phase crucial
• Audio
• Images
• Communication systems
• Linear phase response
• Need FIR filters
• Realizable IIR filters cannot achieve linear
  phase response over all frequencies

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Z-transform Definition

• For discrete-time systems, z-transforms play same
  role as Laplace transforms do for continuous-time
  systems
• Inverse transform requires contour integration
  over closed contour (region) R
• Contour integration covered in a Complex Analysis
  course
• Compute forward and inverse transforms using
  transform pairs and properties

Bilateral Forward z-transform
Bilateral Inverse z-transform
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Common Z-transform Pairs

• hk dk
• Region of convergence entire z-plane
• hk dk-1
• Region of convergence entire z-plane
• hn-1 ? z-1 H(z)

• hk ak uk
• Region of convergence z gt a
• z gt a is the complement of a disk

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Region of Convergence

• Region of the complex z-plane for which forward
  z-transform converges

• Four possibilities (z0 is a special case that
  may or may not be included)

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Stability

• Rule 1 For a causal sequence, poles are inside
  the unit circle (applies to z-transform functions
  that are ratios of two polynomials) OR
• Rule 2 Unit circle is in the region of
  convergence. (For continuous-time signal,
  imaginary axis would be in region of convergence
  of Laplace transform.)
• Example
• Stable if a lt 1 by rule 1 or equivalently
• Stable if a lt 1 by rule 2 because zgta and
  alt1

pole at za
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Transfer Function

• Transfer function is z-transform of impulse
  response e.g. for FIR filter with M taps (slide
  5-3)
• Region of convergence is entire z-plane
• FIR filters are always stable
• Substitute z e j w into transfer function to
  obtain frequency response
• Valid when unit circle is in region of
  convergence (i.e. for stable systems according to
  rule 2 on last slide)