CSE-223

Digital Signal
Processing
DSP
BOOKS
BOOKS
SIGNALS, SYSTEMS, AND
SIGNAL PROCESSING
SIGNALS, SYSTEMS, AND
SIGNAL PROCESSING
 SIGNALS, SYSTEMS, AND
SIGNAL PROCESSING
• A signal is defined as any physical quantity that varies
with time, space, or any other independent variable or
variables.
• Mathematically: a signal is a function of one or more
independent variables. S(t) = A sin (ωt + θ)
• Example (One independent variable)

• Example (Two independent variables)

• An example of a signal that is a function of two

independent variables is an image signal.
 SIGNALS, SYSTEMS, AND
SIGNAL PROCESSING
• There are cases where such a functional relationship is
unknown or too highly complicated to be of any practical
use. For example, a speech signal:
 SIGNALS, SYSTEMS, AND
SIGNAL PROCESSING
• Signal generation is usually
associated with a system that
responds to a stimulus or force.
• In a speech signal, the system
consists of the vocal cords and the
vocal tract, also called the vocal
cavity.

• A system may also be defined as a

physical device (or a software) that
performs an operation on a signal.
• A filter used to reduce the noise
corrupting a signal is an example
of a system .
 SIGNALS, SYSTEMS, AND
SIGNAL PROCESSING
• A system performs an operation on a
signal. In general, the system is
characterized by the type of operation that
it performs on the signal. Those
operations are usually referred to as
signal processing.
• For example a filter eliminates noise from
a signal.

• Signal processing can be performed by using:

• Analog signal processing systems.
• Digital signal processing systems.
 BASIC ELEMENTS OF A DIGITAL SIGNAL
PROCESSING SYSTEM
• Most of the signals encountered in science and
engineering are analog in nature.
• Analog signals are functions of a continuous variable,
such as time or space, and usually take on values in a
continuous range.
• Such signals may be processed directly by appropriate
analog systems.
 BASIC ELEMENTS OF A DIGITAL SIGNAL
PROCESSING SYSTEM
• Digital signal processing systems perform the processing
digitally.
• Digital signal processor may be a –
• large programmable digital computer
• small microprocessor programmed to perform the desired
operations
• hardwired digital processor configured to perform a
specified set of operations
ADVANTAGES OF DIGITAL OVER
ANALOG SIGNAL PROCESSING
Advantages
Digital Signal Processing Analog Signal Processing
Allows flexibility in reconfiguring the Reconfiguration needs redesign of the
digital signal processing operations hardware followed by testing and
simply by changing the program. verification.
Provides much better control of Tolerances in analog circuit
accuracy requirements. components make it extremely difficult
for the system designer to control the
accuracy.
Digital signals can be stored. As a Not possible.
consequence, the signals become
transportable and can be processed
off-line in a remote laboratory.
Allows the implementation of more Very difficult to perform precise
sophisticated signal processing mathematical operations on signals in
algorithms. analog form.
In some cases cheaper. Comparatively costly.
 CLASSIFICATION OF SIGNALS
Multichannel Signals
• Signals are generated by multiple sources or multiple sensors.

• Example:
• ground acceleration due to an earthquake
• electrocardiogram (ECG)
 CLASSIFICATION OF SIGNALS
Multidimensional Signals
• If the signal is a function of a single independent variable, the signal is
called a one-dimensional signal.
• A signal is called M-dimensional if its value is a function of M
independent variables.
 CLASSIFICATION OF SIGNALS
Multidimensional Signals
• A black and white television picture may be represented as I (x, y, t)
since the brightness is a function of time. Hence the TV picture may be
treated as a three-dimensional signal.
• A color TV picture may be described by three intensity functions of the
form Ir (x, y, t), Ig (x, y, t ), a n d Ib (x, y, t), corresponding to the
brightness of the three principal colors (red, green, blue) as functions
of time. Hence the color TV picture is a three-channel, three-
dimensional signal, which can be represented by the vector
 CLASSIFICATION OF SIGNALS
Continuous-Time Signals
• Continuous-time signals are defined for every value of time and
they take on values in the continuous interval (a, b), where a can
be – and b can be .
• Examples:
• Another example:
 CLASSIFICATION OF SIGNALS
Discrete-Time Signals
• Discrete-time signals are defined only at certain specific values of
time. These time instants need not be equidistant, but usually
equidistant.
• An example:
• Other examples:
 CLASSIFICATION OF SIGNALS
Discrete-Time Signals may arise in two ways:
• By selecting values of an analog signal at discrete-time instants.
This process is called sampling.

• By accumulating a
variable over a period
of time.
- The number of
sunspots observed
during an interval of
1 year.
 CLASSIFICATION OF SIGNALS
Continuous-Valued Signals
• If a signal takes on all possible values on a finite or an infinite
range, it is said to be continuous-valued signal.
 CLASSIFICATION OF SIGNALS
• A continuous-time discrete-valued signal.
 CLASSIFICATION OF SIGNALS
Discrete-Valued Signals
• If a signal takes on values from a finite set of possible values, it is
said to be a discrete-valued signal.
 CLASSIFICATION OF SIGNALS
• A continuous-time continuous-valued signal is known as an analog
signal.

• A discrete-time signal having a set of discrete values is called a

digital signal.
 CLASSIFICATION OF SIGNALS
Deterministic Versus Random Signals
• The mathematical analysis and processing of signals requires the
availability of a mathematical description for the signal itself. This
mathematical description, often referred to as the signal model.
• Any signal that can be uniquely described by an explicit
mathematical expression, a table of data, or a well-defined rule is
called deterministic signal.
• all past, present, and future values of the signal are known
precisely, with out any uncertainty.
 CLASSIFICATION OF SIGNALS
Deterministic Versus Random Signals
• There are signals that either cannot
be described to any reasonable
degree of accuracy by explicit
mathematical formulas, or such a
description is too complicated to be of
any practical use. The lack of such a
relationship implies that such signals
evolve in time in an unpredictable
manner. We refer to these signals as
random signals.
 THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS

Continuous-Time Sinusoidal Signals

• The nature of time (continuous or discrete) would affect the
nature of the frequency.
• A simple harmonic oscillation, continuous-time sinusoidal
signal:
 THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS

Continuous-Time Sinusoidal Signals

Characterized by the following properties:
• A1. For every fixed value of the frequency F, xa(t) is periodic.
• A2. Continuous-time sinusoidal signals with distinct (different)
frequencies are themselves distinct.
• A3. Increasing the frequency F results in an increase in the
rate of oscillation of the signal.

• frequency is an inherently positive

physical quantity.
• only for mathematical convenience,
we need to introduce negative
frequencies.
 THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS

Continuous-Time Sinusoidal
Signals
Characterized by the following
properties:
• A1. For every fixed value of the
frequency F, xa(t) is periodic.
• A2. Continuous-time sinusoidal
signals with distinct (different)
frequencies are themselves
The complex signal in the top part of the figure
distinct. (Y, in red color) is given by the sum of two sine
functions (s1 and s2) and two cosine functions
• A3. Increasing the frequency F (c1 and c2), each one with different amplitude
results in an increase in the and frequency values [1]

rate of oscillation of the signal.

[1] Seeber, R., & Ulrici, A. (2016). Analog and digital worlds: Part 1. Signal sampling and Fourier
Transform. ChemTexts, 2(4), 18.
 THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS

Discrete-Time Sinusoidal Signals

• may be expressed as:
• n is the sample number, A is the amplitude, is the frequency
in radians per sample, is the phase in radians.
• and f has dimensions of cycles per sample.
THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS
THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS
 THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS

Discrete-Time Sinusoidal Signals

Discrete-time sinusoids whose frequencies are separated by an integer
multiple of 2 are identical.

• Let, θ = 0
• If A=1 then
x(n) = cos(ω0n)
 THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS

Discrete-Time Sinusoidal Signals

Discrete-time sinusoids whose frequencies are separated by an integer
multiple of 2 are identical.

• x(n) = cos(ω0n)
• For n = 1, the sinusoid is cos(ω0) = cos(π/6) = 0.866 [from last example]
If we add 2π with ω0, cos(ω0+2π) = cos(π/6+2π) = 0.866
• For n = 2, the sinusoid is cos(2ω0) = cos(π/3) = 0.5
If we add 2π with ω0, cos(2(ω0+2π)) = cos(π/3+4π) = 0.5
• And so on …
THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS
THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS
THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS
THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME
AND DISCRETE-TIME SIGNALS
ANALOG-TO-DIGITAL AND DIGITAL-
TO-ANALOG CONVERSION
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
SAMPLING OF ANALOG SIGNALS
 SAMPLING OF ANALOG SIGNALS
• An example of aliasing:
SAMPLING OF ANALOG SIGNALS
THE SAMPLING THEOREM
Given any analog signal, how should we select the sampling period T
or, equivalently, the sampling rate Fs?

Components

Ensure that the signal doesn’t contain significant frequency component

above Fmax .
We know that the highest frequency in an analog signal that can be
unambiguously reconstructed when the signal is sampled at a rate Fs =
THE SAMPLING THEOREM
The sampling theorem specifies the minimum-sampling rate at which a
continuous-time signal needs to be uniformly sampled so that the original
signal can be completely recovered or reconstructed by these samples
alone.

This is usually referred to as Shannon's sampling theorem in the

literature.

If a continuous time signal contains no frequency components higher

than Fmax hz, then it can be completely determined by uniform samples
taken at a rate Fs samples per second where Fs ≥ 2Fmax
or, in term of the sampling period T ≤ 1/2Fmax

The minimum sampling rate allowed by the sampling theorem

(Fs=2Fmax) is called the Nyquist rate.

Ref.: Edmund Lai PhD, BEng, in Practical Digital Signal Processing, Newnes, 2003, Pages
14-49 (Sec. 2.2.1).
 QUANTIZATION OF CONTINUOUS-AMPLITUDE SIGNALS
• The process of converting a
discrete-time continuous-
amplitude signal into a digital
signal by expressing each
sample value as a finite
(instead of an infinite) number
of digits, is called quantization.
 QUANTIZATION OF CONTINUOUS-AMPLITUDE SIGNALS
• The process of converting a discrete-time continuous-amplitude
signal into a digital signal by expressing each sample value as a
finite (instead of an infinite) number of digits, is called quantization.
 QUANTIZATION OF CONTINUOUS-AMPLITUDE SIGNALS
• The error introduced in
representing the
continuous-valued signal
by a finite set of discrete
value levels is called
quantization error or
quantization noise.
 QUANTIZATION OF CONTINUOUS-AMPLITUDE SIGNALS
• Samples x(n)
• Quantized samples xq(n)
• Hence, xq(n) = Q[x(n)]
• Quantization error, eq(n) = xq(n) - x(n)

eq(n
x(n) )
 QUANTIZATION OF CONTINUOUS-AMPLITUDE SIGNALS
• Example:
• Analog exponential signal, xa(t) = 0 .9 t, t ≥ 0
• Sampling frequency, fs = 1 Hz
• Discrete-time signal,
 QUANTIZATION OF CONTINUOUS-AMPLITUDE SIGNALS
• Example:
• Analog exponential signal, xa(t) = 0 .9 t, t ≥ 0
• Sampling frequency, fs = 1 Hz
• Discrete-time signal,
 QUANTIZATION OF CONTINUOUS-AMPLITUDE SIGNALS
• Example:
• Analog exponential signal, xa(t) = 0 .9 t, t ≥ 0
• Sampling frequency, fs = 1 Hz
• Discrete-time signal,
 QUANTIZATION OF CONTINUOUS-AMPLITUDE SIGNALS
• In practice, we can reduce the quantization error to an insignificant
amount by choosing a sufficient number of quantization levels.
• Indeed, quantization is an irreversible or noninvertible process since all
samples in a distance ∆/2 about a certain quantization level are
assigned the same value.
QUANTIZATION OF SINUSOIDAL SIGNALS
xa(t) = A cos(Ω0t)
 QUANTIZATION OF SINUSOIDAL SIGNALS

• If the sampling rate Fs satisfies the sampling theorem,

quantization is the only error in the A /D conversion
process.
• The signal xa(t) is almost linear between quantization
levels.
• Quantization error eq(t) = xa(t) – xq(t)
QUANTIZATION OF SINUSOIDAL SIGNALS
QUANTIZATION OF SINUSOIDAL SIGNALS
 CODING OF QUANTIZED SAMPLES

• Discussed
 DIGITAL-TO-ANALOG CONVERSION

• Discussed
THANK YOU !!!