Day 2 Class Notes

8/3/2019 Day 2 Class Notes

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5/2/2012 Linton University College, Mantin, Malaysia 1

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.

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Day 2 Class Notes