Course: ECE 246/446 - Digital Signal Processing
Meets: GOER 108, TR: 12:30 -1:45
Instructor: Jeff Astheimer
Phone: 624-1582, 624-9778
Office: (ECE LIBRARY)
Office Hours: Tuesday (11:30-12:30) and by appointment
Prerequisites: ECE241 and Matlab programming skills
Books: Discrete - Time Signal Processing (3rd Edition)
by Alan V. Oppenheim and Ronald W. Schafer
Computer - Based Exercises for Signal Processing Using Matlab
by James H. McClellan, et. al.
Exams: Midterm (20%), Final Exam (20%)
Homework: Assignments (30%)
Matlab: Assignments (30%)
Digital Signal Processing
General Introduction (1)
Signal implies the transmission of a message
During transmission the message may undergo
transformations that alters it.
Radio signals are a good example.
The original sounds experience a series of transformations
as they make their way to the ears of the listener.
General Introduction (2)
An abbreviated sequence is :
1. Sound is converted into electrical signals
2. Electrical signals are converted into radio waves
3. Radio waves are reflected (multiple times) by layers
of the ionosphere before reaching the receiving antenna
4. Radio waves are converted back into electricity
5. Electrical signals are transformed into sound
General Introduction (3)
These transformations are illustrated in the figure below :
General Introduction (3)
Corrupting influences may appear at each step
Corruption may occur from background noise
Or, the distortion may be more systematic, as in
electronic circuits or components that have less than
A diagram always helps! In a diagram we can make each step
seem more manageable by representing it as a box.
General Introduction (4)
The transformations listed above may be represented in a
diagram as follows :
Sound to Voltage to Transmission of Radio waves Voltage to
Voltage Radio Waves Radio Waves to Voltage sound
Each box has an input and an output, and the original message
makes it way from one end of the process to the other,
undergoing each transformation in succession.
An early goal of signal processing was to model
these transformations and to add additional boxes to
correct the distortions.
The corrections (often called filters) were typically
implemented as analog electronic circuits.
General Introduction (5)
Ultrasonic Imaging Provides another Example
General Introduction (6)
Transmit at Recieve at
positions x1 , x2 ,… positions y1 , y2 ,…
Reflections to and from a single transducer
are often not sufficient to form a good image,
so signals from multiple transmitters and
receivers must be combined
General Introduction (7)
In medical ultrasound the signals are distorted by
tissues that are encountered en route to the objects
General Introduction (8)
Early conceptions of signal processing have evolved.
The tools of the trade have been extended to
higher dimensional data.
Added complexities of higher dimensions don't appear,
for example, in the transmission of television signals.
Once an image has been rasterized it is essentially one
dimensional and can be treated the same way as
General Introduction (9)
But optical transmission that occurs in image formation
is distinctly different.
Since points of an image can be influenced by neighbors from
every direction, local distortions can be much more complex.
General Introduction (10)
Sometimes interest is in the transformation itself
(The Media is the Message), and the signals are only used
used as probes as in X - rays.
The ratio of the intensity of the output to the intensity of the input
determines the density of the material that was encountered
along each ray.
General Introduction (11)
Modern imaging methods make the old X - ray systems seem
simplistic. Reconstruction of 2 - and 3 - dimensional facsimiles
(via back -projection or Fourier inversions) are common, and
these techniques have become part of the DSP arsenal.
General Introduction (12)
Physical principles that are being employed in sensors, detectors
and transducers (e.g. MRI or PET imaging, etc.) are becoming
more and more sophisticated
Part of the interest and challenge in DSP is to blend the processing
theory with the physical theory of the measurement.
In experimental settings things always goes wrong, and you
need to figure out if the problem is in the physics or in the
General Introduction (13)
Another common DSP task is identification of special features in a signal.
The feature could be as simple as a peak or a voltage transition, but could also
be more complicated, as in character or pattern recognition applications.
This image shows time varying
potentials at different electodes
around the head. When the subject
is presented with a stimulus, a
potential change is evoked at
different times for different
The signals must be processed to
identify the evoked potentials
General Introduction (14)
We have been discussing 'Signal Processing',
What about the word 'Digital'?
x ( nT )
x (t )
x ( ( n + 1) T )
'Digital' indicates concern with signals that
are made up of discretely digitized samples.
Digitization is a critical practical matter.
Signal processing depends on digital electronics.
General Introduction (15)
As signals have proliferated, new processing steps have
have been mandated by practical considerations.
This chart shows FCC frequency band allocations
for the electomagnetic spectrum
General Introduction (16)
Various bands fill up with telephone conversations, cartoon shows,
talk radio etc., making it necessary to find ways to encode voice and other
signals into condensed forms . Thus, compression and decompression algorithms
have become very important
Compress Transmit Decompress
Also, to maintain some semblance of privacy, encryption and decryption
coding has become essential.
Encrypt Transmit Decrypt
Characterization of DSP (1)
These examples are only a small sampling of a very
wide range of areas that digital signal processing has branched
However, the fundamental principles are well established,
and have evolved into an elegant theory.
The next few slides try to identify some of the distinctive
characteristics of this theory.
Characterization of DSP (2)
To begin with, DSP procedures are operations that are applied
to discrete signals.
However, sampling of continuous (analog) signals
(A/D Conversion), and also interpolation of discrete signals
into continuous forms (D/A Conversion), enable the discrete
operations to have a broader range of application.
Continuous A/D Digital Signal D/A Continuous
Input Converter Processing Converter Output
Continuous Signal Processing
Characterization of DSP (3)
The purposes of the DSP operations are varied. Some of the more
important ones are
• Correct Distortion
(de - blurr)
• Signal Decomposition
(separate messages or separate message and noise)
• Feature Enhancement
(boost signal components, sharpen images, etc.)
• Noise Reduction
• Signal Analysis
(transitions, patterns, peaks, frequency distribution, etc.)
• Signal Compression
• Signal Encryption
• Signal Transformation
(Fourier, Walsh, Wavelet)
Characterization of DSP (4)
It may seem like almost any type of processing
can be placed under the DSP heading, but this isn't so.
Many operations (e.g statistical calculations such as the analysis of
variance) belong to altogether different disciplines.
• DSP procedures are distinguished by being closely tied to
the operations of convolution and Fourier decomposition.
That an entire discipline is devoted to such a seemingly limited
set of operations results from a confluence of three factors :
Characterization of DSP (5)
Factor 1. A very large number of naturally occuring and man
made signal influences share two important properties :
1. The influences are time invariant
2. The influences are linear
(satisfy a superposition principle)
These types of influences are called linear time invariant
systems (or simply LTI systems) and are characterized by
Characterization of DSP (6)
Factor 2. Analysis of convolution operators is greatly
facilitated by Fourier (harmonic) decomposition, for
which there is a very large theoretical foundation.
The theory of Fourier analysis is at the heart of many
scientific disciplines (linear differential equations,
potential theory, heat conduction, wave propagation, etc.)
Characterization of DSP (7)
Factor 2. (continued) Many disciplines contribute alternate
models that can help to understand and interpret different
types of LTI systems.
Broad interest in these problems has stimulated development
of the subject.
Characterization of DSP (8)
Factor 3. Despite the preceding factors, DSP would probably
not play such an important role in engineering were it not for
the fact that the principal processing step in all DSP operations
is a simple : multiply and add.
This feature is immediately apparent in the convolution
operation, and can be easily verified for the discrete Fourier
transform as well.
Characterization of DSP (9)
Factor 3. (continued) Furthermore, the Fast Fourier Transform,
which was rediscovered by Cooley and Tukey in the 60's, recasts
the transform calculation as an alterante set of multiply and add
operations that is extremely efficient.
The ability of digital hardware, in the form of either a computer
or a dedicated signal processing component (DSP chip, array
processor, etc.) to perform massive repititions of the multiply
and add operation (either serially or in parallel) means that DSP
can be brought to bear on signals in all kinds of circumstances
and, in particular, in real time.
This is what makes it possible for digital signal processing to be
used in place of the older and cruder approach of using analog
Characterization of DSP (10)
This last factor gives DSP a rather different flavor from
more a straightforward study of Fourier analysis, or
studies of mathematical models in other scientific disciplines.
For although digital hardware may seem to have unlimited
capability, there are always limits, and to accomodate these
limits the subject is infused with a practical aspect.
A standard engineering question is not
'How can this filter be implemented ?'
'How can this filter be implemented
using only 15 multiply and adds for each input sample?'
Course Outline (1)
The text for the course is
Discrete - Time Signal Processing (3rd Edition)
by Alan V. Oppenheim and Ronald W. Schafer
We will try to cover the first 8 chapters.
Course Outline (2)
Chapter 2 : Discrete - Time Signals and Systems
• Discrete - time sequences are introduced and
their basic operations and features are described
• The notion of a system as an operator is explained
and LTI characteristics are distinguished.
• LTI systems can sometimes be characterized a solutions
to a linear constant coefficient difference equation (LCCDE).
This characterization will have special importance for the
practical implementation of real time systems.
• The Discrete - Time Fourier Transform is defined and important
properties of the transform are identified
Course Outline (3)
Chapter 3 : The z - transform
• The z - transform is an extension of the discrete - time Fourier
transform to regions of the complex plain. The importance of
this transform won't be fully appreciated until chapter 5.
• Important properties of the transform are identified.
• Methods of computing the z - transform are discussed, with
special emphasis placed on determining the z - transform of
associated with LCCDE's
Course Outline (4)
Chapter 4 : Sampling of Continuous - Time Signals
• This chapter examines A/D and D/A transformations,
and characterizes these operations in both the time
and frequency domain. This leads to the fundamental
notion of the nyquist sampling rate.
• The D/A operation provides a natural way to associate a
continuous signal to any discrete sequence, and this
association is often exploited to apply continuous
operations into discrete sequences.
• An example of particular importance is the operation of
changing the sampling rate of a discrete - time signal.
Course Outline (5)
Chapter 5 : Transform Analysis of LTI systems
• The best way to understand a LTI system is to study the frequency
response function of the system (the Fourier transform of the
impulse response). The chapter begins with a discussion of some
basic characteristics of the frequency response function.
• The role of the z - transform is not to help understand systems, but
rather to help design them. The placement of poles and zeros of
the z - transform of the impulse response of LCCDE systems dictates
the behavior of the frequency response.
• Special configurations of poles and zeros lead to systems with special
properties : stable, causal, all -pass, minimum -phase, linear phase
Course Outline (6)
Chapter 6 : Discrete Time Structures
• These structures refer to different schemes for implementing
linear systems that derive from LCCDE's. The purpose of the
structures is to enable systematic exploration of different
implementations to optimize their efficiency and numerical
• Diagrammatic representation of different implementations
are described and categorized.
• Methods for determining the numerical error associated with
different diagrams are explored.
Course Outline (7)
Chapter 7 : Filter Design Techniques
• A filter is just a LTI system with special frequency response
characteristics. But real time applications often require the
filter to be implemented as an LCCDE system, which means
that it may not be possible to attain the ideal (i.e. exact)
• Filter design entails choosing the LCCDE coefficients to
approximate the frequency response in an optimal manner
Course Outline (8)
Chapter 8 : The Discrete Fourier Transform
• The Discrete Fourier transform is distinct from the
Discrete - Time Fourier transform described in chapter 2
because it only applies to discrete sequences that are
• The effect of this simplification is that the frequency
response function also becomes a periodic discrete
• This has far reaching effects on prior methods, and
offers a powerful alternative approach to system analysis
and design, but this alternative is not as amenable to real
• Collaboration : Working together has definite benefits and
can make homework assignments much more palatable.
However, it muddles the significance of the homework grade,
and can diminish the efficacy of homework as a learning device.
The recommended policy is to only collaborate judiciously.
• Clarity : An important part of the grading for both
homework and matlab assignments will be clarity of
This means, in particular, that matlab code should be
well organized and well documented (lots of comments)