1. Finite Impulse Response Filters

2. Discrete-Time Impulse Signal
Let d[k] 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)
d[k]
h[k]

3. Discrete-time Tapped Delay Line
Impulse response h[k] of finite extent k = 0,…, M-1
Block diagram (finite impulse response filter)
Discrete-time convolution
z-1 …
x[k] S
y[k]
h[0]
h[1]
h[2]
h[M-1]
x[k-1]
Applications of continuous-time tapped delay lines?

4. Discrete-time Convolution Derivation
Output y[k] for input x[k]
Any signal can be decomposed into sum of discrete impulses
Apply linear properties
Apply shift-invariance
Apply change of variables k
h[k]
Averaging filter impulse response 1 2 3
y[k] = h[0] x[k] + h[1] x[k-1]
= ( x[k] + x[k-1] ) / 2

5. 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

6. Convolution Demos Johns Hopkins University Demonstrations(
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) t 1
x(t) Ts
2Ts
h(t) * =
y(t)

7. Linear Time-Invariant Systems
Complex exponentials zk have a special property when they are input into LTI systems
Output will be same complex exponential 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

8. 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 x[k] = e j W k,
H(W) is discrete-time Fourier transform of h[k] H(W) is also called the frequency response
H()

9. 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

10. Example: Ideal Delay Continuous Time Discrete Time Delay by T seconds
Impulse response
Frequency response
Discrete Time
Delay by 1 sample
Impulse response
Frequency response
x(t)
y(t)
x[k]
y[k]
w = W T
Allpass Filter
Linear Phase

11. 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)|
Linear phase
stopband
stopband W W
-Ws
-Wp Wp Ws
passband

12. 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?

13. Mandrill Demo (DSP First)
Five-tap discrete-time averaging FIR filter with input x[k] and output y[k]
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 (sharpens input signal)
Impulse response is {1, -1}
h[k]
First-order difference impulse response k

14. Mandrill Demo (DSP First)
DSP First demos:
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

15. 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 

16. Finite Impulse Response Filters
Duration of impulse response h[k] 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 y[k] 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?

17. 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 h[0] = h[2]
Implementation savings
Reduce number of multiplications from M to M/2 for even-length and to (M+1)/2 for odd-length impulse responses
Reduce storage of impulse response by same amount
TI TMS320C54 DSP has an accelerator instructor FIRS to compute h[0] ( x[k] + x[k-2] ) in one instruction cycle
On most DSPs, no accelerator instruction is available

18. 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
Transition band
Stopband
Red region is forbidden
dp passband ripple ds stopband ripple

19. 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

20. 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

21. Common Z-transform Pairs
h[k] = d[k]
Region of convergence: entire z-plane
h[k] = d[k-1]
h[n-1]  z-1 H(z)
h[k] = ak u[k]
Region of convergence: |z| > |a|
|z| > |a| is the complement of a disk

22. Region of Convergence
Region of the complex z-plane for which forward z-transform converges
Four possibilities (z=0 is a special case that may or may not be included)
Im{z}
Re{z}
Entire plane
Im{z}
Re{z}
Disk
Im{z}
Re{z}
Complement of a disk
Im{z}
Re{z}
Intersection of a disk and complement of a disk

23. 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| < 1 by rule #1 or equivalently
Stable if |a| < 1 by rule #2 because |z|>|a| and |a|<1
pole at z=a

24. 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)