day 2 class notes
TRANSCRIPT
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Module Name/ Module code
Digital signal Processing / EE 3002
CLASS 2
Class: B.Eng ( Hons)
Module leader : CH. Kranthi Rekha
Lecturer, SOEE
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Discrete time systems
Discrete-time system is a device or algorithm that operates on a
discrete-time signal, called the input or excitation, according to some
well-defined rule, to produce another discrete-time signal called the
output or response.
A discrete-time system is defined as a transformation or mappingoperator that maps an input signal x (n) to an output signal y (n).
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Types of Systems
Static and dynamic systems
Causal and non-causal systems
Linear and nonlinear systems
Shift variant and shift invariant systems
Stable and unstable systems
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Memoryless systems
A system is Memoryless if the output y [n] at any instant n depends only on
x [n] at the same n, but no past or future samples of the input.
For example, y [n]= (x[n/2])2 is Memoryless,
But the ideal delay
y [n]= x [n n d] is not a Memoryless system unless n d = 0.
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Linear systems
A system is linear if the principle of superposition applies. Thus if
y1 [n] is the response of the system to the input x1 [n], and y2 [n] the
response to x2 [n], then linearity implies
Additivity:
Scaling:
These properties combine to form the general principle ofsuperposition
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Examples ofLinear Systems
Wave propagation such as sound and electromagnetic waves
Electrical circuits composed of resistors, capacitors, and inductors
Electronic circuits, such as amplifiers and filters
Mechanical motion from the interaction of masses, springs, and dashpots
(dampeners)
Systems described by differential equations such as resistor-capacitor-inductor
networks
Multiplication by a constant, that is, amplification or attenuation of the signal
Signal changes, such as echoes, resonances, and image blurring
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Examples ofLinear Systems(contd)
The unity system where the output is always equal to the input
The null system where the output is always equal to the zero, regardless of the input
Differentiation and integration, and the analogous operations offirst difference and
running sum for discrete signals
Small perturbations in an otherwise nonlinear system, for instance, a small signal being
amplified by a properly biased transistor
Convolution, a mathematical operation where each value in the output is expressed as
the sum of values in the input multiplied by a set of weighing coefficients.
Recursion, a technique similar to convolution, except previously calculated values in the
output are used in addition to values from the input
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Examples ofNonlinear Systems
Systems that do not have static linearity,
Systems that do not have sinusoidal fidelity, such as electronics circuits for: peak
detection, squaring, sine wave to square wave conversion, frequency doubling, etc.
Common electronic distortion, such as clipping, crossover distortion and slewing
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Examples ofNonlinear Systems(contd.)
Multiplication of one signal by another signal, such as in amplitude modulation and
automatic gain controls
Hysteresis phenomena, such as magnetic flux density versus magnetic intensity in
iron, or mechanical stress versus strain in vulcanized rubber
Saturation, such as electronic amplifiers and transformers driven too hard Systems
with a threshold, for example, digital logic gates, or seismic vibrations that are
strong enough to pulverize the intervening rock
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Time-invariant systems
A system is time invariant if a time shift or delay of the input
sequence causes a corresponding shift in the output sequence. That
is, if y [n] is the response to x [n], then y [n- n0] is the response to
x [n - n0].
For example, the accumulator system is time invariant,
But the compressor system for M a positive integer (which selects
every M th sample from a sequence) is time variant.
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Causality
A system is causal if the output at n depends only on the input at n and
past input values for n (n-1,n-2,n-3,.) but not on future inputvalues of n (such as n+1, n+2, n+3,.).
For example, the backward difference system is causal,
but the forward difference system is not causal
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Stability
A system is stable if every bounded input sequenceproduces a bounded output sequence:
For example, the accumulator is an example of anunboundedsystem, since its response to the unit step u [n] is
which has no finite upper bound
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Linear time-invariant systems
If the linearity property is combined with the representation of a
general sequence as a linear combination of delayed impulses, then it
follows that a linear time-invariant (LTI) system can be completely
characterized by its impulse response.
Suppose hk
(n) is the response of a linear system to the impulse (n-k)
at n = k. Since
The principle of superposition means that
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If the system is additionally time invariant, then the response
to (n-k) is h(n-k). The previous equation then becomes
This expression is called the convolution sum. Therefore, an
LTI system has the property that given h(n), we can find y(n)
for any input x(n). Alternatively, y(n) is the convolution of
x(n) with h(n), denoted as follows:
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Representation of an arbitrary sequence
Any arbitrary sequence x (n) can be represented in terms of delayed and
scaled sequence (n).
Let x (n) be an infinite sequence as shown:
That is, x (n) is thus given as
Where (n - k) is unity for n = k and zero for all other terms.
-3 -2 -1 0 1 2 3
0.5
1.5 1.5
0.5
1
1
------ ------
1.5
k
knkxnx )(
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Impulse response and convolution
A discrete time system performs an operation on an input signal based on a
predefined criteria to produce output signal.
Let the input signal x (n) is the system excitation, and y (n) is the system
response. This is shown as:
If the input to the system is a unit impulse i.e., x (n) = (n) then the output of
the system is known as impulse response denoted by h (n) where h (n) = T [
(n)].
Now the system response becomes
Which equals
k
knTkxny
Tx (n) y (n) = T x (n)
k
knhkxny
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Thus from the above we can say that a linear time invariant system, if
the sequence x (n) and impulse response h (n) are given, we can find
the output y (n) by using the equation
Which is known as the convolution sum and can be represented
y (n) = x (n) * h (n)
k
knhkxny
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Procedure to find convolution sum of two sequences
Step 1: Choose an initial value of n, the starting time for evaluating the output
sequence y (n). If x (n) starts at n =n1 and h (n) starts at n = n2 then
n = n1 + n2-1
Step 2: Express both sequences in terms of the index k.
Step 3: Fold h (k) about k =0 to obtain h (-k) and shift n to right by one sampleand do step 4.
Step 4: Multiply the two sequences x (k) and h (n - k) element by element and
sum up the products to get y (n).
Step 5: Increment the index n, shift the sequence h (n - k) to right by one
sample and step 4.
Step 6: Repeat step 5 until the sum of products is zero for all the remaining
values of n.