DiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
Lesson Title: FIR Frequency Response
Lesson Number: 16 (Section 67 to 69)
Background:
In Chapter 6, the concept of frequency response for FIRs was introduced.
Sections 67 and 68
basically beat it to death beginning with the now too familiar MA filter (a.k.a., running average
filter).
MA Filter
At the end of Chapter 6, the steadystate frequency response of a discretetime MA FIR system
is analyzed.
The inputoutput relationship of an L
th
order linear phase MA FIR is given by.
[
]
[
]
∑

=

=
1
0
1
L
k
k
n
x
L
n
y
1.
The frequency response is given by (see previous lessons or textbook for derivation):
(
29
(
29
(
29
(
29
(
29
(
29
2
/
1
2
/
1
1
0
2
/
sin
2
/
sin
1





=
=
=
=
∑
L
j
j
L
L
j
L
k
j
j
e
e
D
e
L
L
e
L
e
H
ω
2.
where
ω
is a normalized frequency and D
L
is called the Dirichlet function (MATLAB
diric
).
D
L
(e
j
ω
) is a real number and MA’s phase response is the linear 
ω
(L1)/2 with a group delay of
τ
= (L1)/2.
Example
The Dirchlet function for L=11 is displayed in Figure 1.
>> t=0:1/100:2*pi;
>> tt=pi+t;
>> d=diric(tt,11);
>> plot(d)
The corresponding phase slope would be
τ
=5 and the phase response is 5
ω
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentDiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
Figure 1:
Dirichlet function for L=11.
The authors spend a few moments discussing the obvious properties of the Dirchlet function
(read them) and then embark on study of image processing.
Survey of 2D FIR filtering
Image processing is an important element of signal processing. The engineer is often
responsible for processing 2D and 3D images in order to alter or enhance an image in a
manner that will accentuate or suppress image attributes or human interpretability.
Digital images are represented by a set of pixels (
viz
; samples) that are separated in space.
Spatial domain
image processing refers to the direct manipulation of a set of those pixels that
compose an image.
Spatial images are sampled in terms of pixels/meter metrics instead of
samples/second (Sa/s).
While spatial domain processing involves procedures that operate
directly on pixels, spatial frequency domain image processing involves the use of a Fourier
transform or its equivalent. In the spatial domain, harmonics are defined in terms of a
fundamental period measured in spatial coordinates (e.g., meters) rather than seconds as is the
case for temporal signals. For example, a 1000point DFT of an image being sampled at 1
pixels/mm, has a fundamental period of 1000 mm or 1 meter. The fundamental spatial
frequency is then 1 cycle/m.
Consider a 256x256 8bit B&W image of Albert (see Figure 2).
The
following Matlab code can be used to import, display, and transform
an image.
EDU>> image=imread('image_name.bmp','bmp');
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 ?
 Image processing, Frequency, discretetime signals, Dr. Fred J. Taylor, Dr. Fred J., Fred J. Taylor

Click to edit the document details