The document discusses the applications and principles of digital signal processing (DSP) across various fields, including audio processing, telecommunications, and biomedical engineering. It covers essential topics like signals and systems, discrete-time signal analysis, digital filters, and the importance of the Nyquist sampling theorem. Key concepts such as deterministic vs. random signals, frequency in discrete-time signals, and waveform properties are also highlighted.

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Digital Signal Processing
Dr. Rakesh R Warier

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Applications of Digital Signal Processing
• Audio and speech processing,
• Sonar, radar and other sensor array processing,
• Spectral density estimation,
• Digital image processing,
• Data compression,
• Signal processing for telecommunications,
• Control systems,
• Biomedical engineering,
• and Seismology, among others

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Contents
• Review of Signals and Systems
• Analysis and representation of discrete-time signal systems, including
discrete-time convolution,
• Difference equations, the z-transform, and the discrete-time Fourier
transform. Emphasis is placed on the similarities and distinctions
between discrete-time,
• Digital filters, (FIR and IIR),
• Fast Fourier transform algorithm.

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The most important algorithm!

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Review of signals and systems
• What is a signal?
• What is a system?
• Analog vs Digital

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Signals
• Continuous vs Discrete Signals
• A signal x(t) is analog and
continuous-time if both x and t
are continuous variables (infinite
resolution). Most real world
signals are analog and
continuous-time.
• Dependent vs independent
variable

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Examples Continued
12 lead ECG signal

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Discrete Signals
• Representations
• Bracket notation
• Stem plot

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Digital vs Discrete-time
• A signal x[n] is analog and discrete-
time if the values of x are continuous
but time n is discrete (integer-valued).
• A signal x[n] is digital and discrete-time
if the values of x are discrete (i.e.,
quantized) and time n also is discrete
(integer-valued). Computers store and
process digital discrete-time signals.
Not covered in this course.

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Deterministic vs Random Signals
• Any signal that can be represented by
a unique model (formula, table or
mathematical expression) without any
uncertainty is called deterministic
signal.
• In many practical applications, there
are signals that cannot be described
by any reasonable accuracy by any
explicit mathematical formula, and
they evolve in an unpredictable
manner and are called random signals.

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Continuous time Sinusoids
• Continuous time sinusoids with
different frequencies are distinct.
• Increasing frequency results in
increasing oscillations

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Periodic Signals (continuous)
• A periodic signal of period satisfies
the periodicity property:
for all integer values of n and all times
t.

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Frequency (discrete time signal)
• A discrete signal is periodic with period N>0 if
, for all
Smallest value for which this is true is called the fundamental
period.

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Example
• Find the frequency (=50)
• Solve for N

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Example
• Find the frequency ()
• Solve for N

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Example
• Find the frequency ()
• Solve for N
• A discrete-time sinusoid is periodic only if the frequency is a rational
number!

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Frequency in discrete-time signals

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Waveform Properties- Symmetry
• A signal x(t) exhibits even symmetry
if its waveform is symmetrical with
respect to the vertical axis.
• The shape of the waveform on the
left-hand side of the vertical axis is
the mirror image of the waveform
on the right-hand side

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Waveform Properties – Odd Symmetry
• Odd Symmetry
• A signal exhibits odd symmetry if
the shape of its waveform on the
left-hand side of the vertical axis is
the inverted mirror image of the
waveform on the right-hand side.

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Splitting Odd and Even Components

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Fourier Formula

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Fourier Series

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Time – Frequency Duality

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Fourier Series

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Sampling
• Consider a sinusoidal signal
x(t) = cos(2π1000t),
sampled at fs = 8000 samples per
second.
• The sampling interval is Ts = 1/8000 s,

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Sampling Theorem
• Let x(t) be a real-valued, continuous-time,
low pass signal bandlimited to a maximum
frequency of B Hz.
• Let x[n] = x(nTs) be the sequence of numbers
obtained by sampling x(t) at a sampling rate
of fs samples per second, that is, every Ts =
1/fs seconds.
• Then x(t) can be uniquely reconstructed from
its samples x[n] if and only if fs > 2B.
• The sampling rate must exceed double the
bandwidth.
• The minimum sampling rate 2B is called the
Nyquist sampling rate.

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Aliasing
• The sampling rate must exceed
double the bandwidth.
• The minimum sampling rate 2B is
called the Nyquist sampling rate.

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Practice Problem

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Solution

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Sampling at Nyquist rate

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Homework

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References
• Digital Signal Processing: Principles, Algorithms, and Applications,
John G. Proakis. Northeastern University. Dimitris G. Manolakis.
Boston College, 4th
ed.