Download Digital Signal Processing

Digital signal Processing
By
Dileep Kumar
[email protected]
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Lecture 1-2
Basic Concepts
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• Signal:
A signal is defined as a function of one or more variables
which conveys information on the nature of a physical
phenomenon. The value of the function can be a real
valued scalar quantity, a complex valued quantity, or
perhaps a vector.
• System:
A system is defined as an entity that manipulates one or
more signals to accomplish a function, thereby yielding
new signals.
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• Continuos-Time Signal:
A signal x(t) is said to be a continuous time signal if it is
defined for all time t.
• Discrete-Time Signal:
A discrete time signal x[nT] has values specified only at
discrete points in time.
• Signal Processing:
A system characterized by the type of operation that it
performs on the signal. For example, if the operation is
linear, the system is called linear. If the operation is nonlinear, the system is said to be non-linear, and so forth.
Such operations are usually referred to as “Signal
Processing”.
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Basic Elements of a Signal Processing
System
Analog input
signal
Analog output
signal
Analog
Signal Processor
Analog Signal Processing
Analog
input
signal
A/D
converter
Digital
Signal Processor
D/A
converter
Analog
output
signal
Digital Signal Processing
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• Advantages of Digital over Analogue Signal
Processing:
A digital programmable system allows flexibility in
reconfiguring the DSP operations simply by changing the
program. Reconfiguration of an analogue system usually
implies a redesign of hardware, testing and verification
that it operates properly.
DSP provides better control of accuracy requirements.
Digital signals are easily stored on magnetic media (tape
or disk).
The DSP allows for the implementation of more
sophisticated signal processing algorithms.
In some cases a digital implementation of the signal
processing system is cheaper than its analogue
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counterpart.
DSP Applications
Space
Medical
Space photograph enhancement
Data compression
Intelligent sensory analysis
Medical image storage and retrieval
Image and sound compression for
Commercial multimedia presentation.
Movie special effects
Video conference calling
Telephone
Video and data compression
echo reduction
signal multiplexing
filtering
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Classification of Signals
•Deterministic Signals
A deterministic signal behaves in a fixed known way with
respect to time. Thus, it can be modeled by a known
function of time t for continuous time signals, or a known
function of a sampler number n, and sampling spacing T
for discrete time signals.
• Random or Stochastic Signals:
In many practical situations, 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.
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Even and Odd Signals
A continuous time signal x(t) is said to an even signal if it
satisfies the condition
x(-t) = x(t) for all t
The signal x(t) is said to be an odd signal if it satisfies the
condition
x(-t) = -x(t)
In other words, even signals are symmetric about the
vertical axis or time origin, whereas odd signals are
antisymmetric about the time origin. Similar remarks
apply to discrete-time signals.
Example:
even
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odd
odd
Periodic Signals
A continuous signal x(t) is periodic if and only if there
exists a T > 0 such that
x(t + T) = x(t)
where T is the period of the signal in units of time.
f = 1/T is the frequency of the signal in Hz. W = 2/T is the
angular frequency in radians per second.
The discrete time signal x[nT] is periodic if and only if
there exists an N > 0 such that
x[nT + N] = x[nT]
where N is the period of the signal in number of sample
spacings.
Example:
Frequency = 5 Hz or 10 rad/s
0
0.2
0.4
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Continuous Time Sinusoidal Signals
A simple harmonic oscillation is mathematically
described as
x(t) = Acos(wt + )
This signal is completely characterized by three
parameters:
A = amplitude, w = 2f = frequency in rad/s, and  =
phase in radians.
A
T=1/f
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Discrete Time Sinusoidal Signals
A discrete time sinusoidal signal may be expressed as
x[n] = Acos(wn + )
- < n < 
Properties:
• A discrete time sinusoid is periodic only if its frequency is a rational
number.
• Discrete time sinusoids whose frequencies are separated by
an integer multiple of 2 are identical.
• The highest rate of oscillation in a discrete time sinusoid is
attained when w =  ( or w = - ), or equivalently f = 1/2 (or f = 1/2).
1
0
-1
0
2
4
6
8
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Energy and Power Signals
•A signal is referred to as an energy signal, if and only if
the total energy of the signal satisfies the condition
0<E<
•On the other hand, it is referred to as a power signal, if
and only if the average power of the signal satisfies the
condition
0 < P< 
•An energy signal has zero average power, whereas a power
signal has infinite energy.
•Periodic signals and random signals are usually viewed as
power signals, whereas signals that are both deterministic and
non-periodic are energy signals.
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Basic Operations on Signals
(a) Operations performed on dependent
variables
1. Amplitude Scaling:
let x(t) denote a continuous time signal. The signal y(t)
resulting from amplitude scaling applied to x(t) is
defined by
y(t) = cx(t)
where c is the scale factor.
In a similar manner to the above equation, for discrete
time signals we write
2x(t)
y[nT] = cx[nT]
x(t)
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2. Addition:
Let x1 [n] and x2[n] denote a pair of discrete time signals.
The signal y[n] obtained by the addition of x1[n] + x2[n]
is defined as
y[n] = x1[n] + x2[n]
Example: audio mixer
3. Multiplication:
Let x1[n] and x2[n] denote a pair of discrete-time signals.
The signal y[n] resulting from the multiplication of the
x1[n] and x2[n] is defined by
y[n] = x1[n].x2[n]
Example: AM Radio Signal
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(b) Operations performed on independent
variable
• Time Scaling:
Let y(t) is a compressed version of x(t). The signal y(t)
obtained by scaling the independent variable, time t, by
a factor k is defined by
y(t) = x(kt)
– if k > 1, the signal y(t) is a compressed version of
x(t).
– If, on the other hand, 0 < k < 1, the signal y(t) is an
expanded (stretched) version of x(t).
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Example of time scaling
1
0.9
0.8
0.7
Expansion and compression of the signal
e-t.
exp(-t)
0.6
0.5
exp(-2t)
0.4
exp(-0.5t)
0.3
0.2
0.1
0
0
5
10
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Time scaling of discrete time systems
x[n]
10
5
-2
-1
0
1
2
3
0
-1.5
5
-1
-0.5
0
0.5
1
1.5
0
-6
-4
-2
0
n
2
4
6
5
x[2n]
x[0.5n]
0
-3
10
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Time Reversal
• This operation reflects the signal about t = 0
and thus reverses the signal on the time scale.
x[n]
5
1
2
-5
0
3
4
5
3
4
5
n
x[-n]
0
0
0
1
2
n
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Time Shift
x[n-3]
x[n+3]
x[n]
A signal may be shifted in time by replacing the
independent variable n by n-k, where k is an
integer. If k is a positive integer, the time shift
results in a delay of the signal by k units of time. If
k is a negative integer, the time shift results in an
advance of the signal by |k| units in time.
1
0.5
0 -2
1
0.5
0 -2
1
0.5
0 -2
0
2
4
6
8
10
0
2
4
6
8
10
0
2
n4
6
8
10
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