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UNIT-I

DISCRETE TIME SIGNALS AND SYSTEMS

Sampling:

Sampling is the process of recording the values of a signal at given points in time.

For A/D converters, these points in time are equidistant. The number of samples taken

during one second is the called sample rate. Keep in mind that these samples are still

analogue values. The mathematic description of the ideal sampling is the multiplication

of the signal with a sequence of Dirac pulses. In real A/D converters the sampling is

carried out by a sample-and-hold buffer. The sample-and-hold buffer splits the sample

period in a sample time and a hold time. In case of a voltage being sampled, a capacitor is

switched to the input line during the sample time. During the hold time it is detached

from the line and keeps its voltage.

Quantization:

Quantization is the process of representing the analog voltage from the sample-

and-hold circuit by a fixed number of bits. The input analog voltage (or current) is

compared to a set of pre-defined voltage (or current) levels. Each of the levels is

represented by a unique binary number, and the binary number that corresponds to the

level that is closest to the analog voltage is chosen to represent that sample. This process

rounds the analog voltage to the nearest level, which means that the digital representation

is an approximation to the analog voltage. There are a few methods for quantizing

samples. The most commonly used ones are the dual slope method and the successive

approximation.

Aliasing is a common problem in digital media processing applications. Many readers

have have heard of "anti-aliasing" features in high-quality video cards. This page willexplain what Aliasing is, and how it can by avoided.

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Aliasing:

Aliasing is a effect of violation of the Nyquist-Shannon-Sampling-Theory. During

the sampling the base band spectrum of the sampled signal ist mirrored to every multifold

of the sampling frequency. These mirrored spectra are called alias. If the signal spectrum

reaches farther than half the sampling frequency base band spectrum and aliases touch

each other and the base band spectrum gets superimposed by the first alias spectrum. The

easiest way to prevent aliasing is the application of a steep sloped low-pass filter with

half the sampling frequency before the conversion.

Anti-aliasing:

The sampling rate for an analog signal must be at least two times as high as the

highest frequency in the analog signal in order to avoid aliasing. Conversely, for a fixed

sampling rate, the highest frequency in the analog signal can be no higher than one half

of the sampling rate. Any part of the signal or noise that is higher than one half of the

sampling rate will cause aliasing. In order to avoid this problem, the analog signal is

usually filtered by a low pass filter prior to being sampled, and this filter is called an anti-

aliasing filter. Sometimes the reconstruction filter after a digital-to-analog converter is

also called an anti-aliasing filter.

Signals:

Any physical quantity that varies with time, space or any other independent

variables is called a signal. Contain information about the behavior or nature of some

phenomenon. Signals are the link up between systems. Signal is a record of an activity

containing information.

Ex: x(t) = sin(3t)

x[n] = 10 cos[5n]

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Types of Signals:

1. Multichannel Signal:

Signals generated by multiple sources or multiple sensors. Ex: EEG signal.

2. One dimensional Signals:

A function of single independent variable. Ex: v(t) = Vm sin(t)3. Multi Dimensional Signals:

A function of two or more independent variables.

Ex: Photo 2 dimensional signal

Ex: B/W TV Picture intensity 3 dimensional signal, I(x,y,t)

Continuous time and Discrete time signals:

Continuous-time Signal x(t):

It is also called analogue signal.

It is continuous in its time axis.

Discrete-time Signal x[n]:

The time axis is discrete.

Independent variable (usually time) has integer values.

It is sequence of numbers. e.g. x[n]=[,3,8,9,0,-1,]

t

Amplitude

nT

Amplitude

-T T

2T

3T 4T

5T 6T 7T

0

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DISCRETE TIME SIGNALS:

The discrete time signals are obtained by time sampling of continuous time

signals.

It can be defined at only at sampling instants.

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Types of Discrete time Signals

1. Deterministic & Non-deterministic

2. Periodic & Aperiodic

3. Even & Odd

4. Energy & PowerDeterministic & Non-deterministic:

Deterministic:

Functions that are completely specified in time.

The nature and amplitude at any time can be predicted.

The pattern is regular and can be characterized mathematically.

Ex: x[n] = {1,0,3,-8}

Non-deterministic (Random):

Signals whose occurrence is random in nature.

Pattern is quite irregular.

The amplitude at any time can not be predicted in advance.

Ex. Noises

Explanation:

Deterministic and Random Signals: A deterministic signal is a signal about which there is no uncertainty with

respect to its value at any time.

A deterministic signal can be completely represented by mathematical equation at

any time.

For example sine wave, cosine wave, triangular, exponential pulse etc are

deterministic signals, since they have unique mathematical equations.

An example of deterministic signal is,

i.e x(t) = A Sin t

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A random signal has some degree of uncertainty before it actually occurs.

That means random signal cannot be defined by some mathematical

expression.

The value of random signal is not predefined or it cannot be calculated from

previous values of the signal.

Random signals are broadly called as Noise signals.

An example of random signal is,

Periodic & Aperiodic:

Periodic:

Should exhibit periodicity.

x[n+N] = x[n], - < n < , where N is the period.

Aperiodic:

Does not satisfy the above condition.

Explanation:

Periodic and Non-periodic Signals:

A signal is said to be periodic if it repeats after a fixed time period.

This can be defined by mathematical expression as,

X(t) = x (t + T0)

An example of periodic signal is sine wave.

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This waveform can be represented by the following mathematical equation

i.e.,

= x (t)

A signal is said to be non-periodic if it does not repeat.

Some times non-periodic signals are said to have a period T0 equal to

infinity.

An example of non-periodic signal is decaying exponential pulse.

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Even & Odd:Even:

Exhibits symmetry in the time domain.

The signal must be identical to its reflection about the origin.

x[n] = x[-n]

Odd:

Exhibits anti-symmetry.

The signal must be identical to the negative of its reflection about the origin.

x[n] = -x[-n]

Explanation:

Symmetric and Non-symmetric Signals or Even and Odd Signals:

Odd or nonsymmetrical signal:

A signal x(t) is said to be odd or non-symmetric if,

x(-t) = -x(t)

This means if the sign of time axis is changed, it also changes the sign of amplitude.

For example consider the sine wave starting at zero. This sine wave is shown.

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= -x(t)

Even or symmetric signal:

A signal is said to be even or symmetric if,

x(-t) = x(t)

This means even if the sign of the time axis is changed, the sign of the amplitude is not

changed.

For example consider the cosine wave as shown in Fig. below.

= A cos (2ft) Since cos (-) = cos

= x (t)

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Energy & Power:

Let x[n] be a discrete-time signal

The energy content of the signal x[n] is as follows:

The power content of the signal is:

If E is finite and P = 0, then x[n] is Energy signal.

If P is finite and E = infinite, then x[n] is Power signal.

=

=n

nxE2

][

+

= +

=

N

NnN

nxN

P2

][12

1lim

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Singularity Functions

These functions are important classifications of aperiodic signals. They are used to

represent more complicated signals.

Types:

Unit impulse function / Delta function

Unit step function

Unit ramp function

Delta function is the basic singularity function and other singularity functions are

derived by repeated integration or differentiation of the delta function.

Unit impulse function

[n] = 1, n = 0

0, n 0

Unit step function

u[n] = 0, n < 0

1, n 0

1

-3 -2 -1 0 1 2 3

1

-3 -2 -1 0 1 2 3

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Unit ramp function

r[n] = 0, n < 0

n, n 0

Representation of discrete time signals

Graphical representation

Functional representation

Tabular representation

Sequence representation

1

-3 -2 -1 0 1 2 3

2

3

n

x[n]

X[n] =1, n = 0,4,75, n = 3-4, n = 1,2,5,6

n

x[n]

0 1 2 3 4 5 6 7

1 -4 -4 5 1 -4 -4 1X[n] = {1, -4, -4, 5, 1, -4, -4, 1}

Graphical

Functional

Tabular Sequence

1

2

3

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Operations on discrete time signals

Shifting:

y[n] = x[n-k]

x[n] = input, y[n] = outputk = +ive, delay the sequence - right shift

k = -ive, advance the sequence left shift

Time reversal:

y[n] = x[-n]

Shifting after time reversal :

x[-n+2] is x[-n] delayed by 2 units

x[-n-2] is x[-n] advanced by 2 units

Time scaling:

y[n] = x[n]

< 1, up sampling

> 1, down sampling

Scalar multiplication:

y[n] = a x[n]

Signal multiplier:

y[n] = x1[n] x2[n]

X[n] a y[n]

Xx1[n]x1[n]

x2[n]x2[n]

y[n]

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Signal addition:

y[n] = x1[n] + x2[n]

Discrete Time Systems

Definition:

Types:

1. Static & Dynamic systems

2. Time-variant & Time-invariant systems

3. Causal & Non-causal systems

4. Linear & Non-linear systems

5. Stable & Unstable systems

6. FIR & IIR systems

Static & Dynamic:

Static:

The output at any instant depends on the input samples at the same time, but not

on the past or future samples of the input.

Memoryless system.

Ex: y[n] = a x[n] and y[n] = a x2[n]

Dynamic: System with memory.

Ex: y[n] = x[n-1] + x[n-2]

+x1[n]x1[n]

x2[n]x2[n]

y[n]

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Time variant & Time invariant:

Time invariant:

o The input output characteristics do not change with time.

o If y[n] = H{x[n]} then, y[n-k] for an input of x[n-k] will be y[n], time shifted by k

units.

o The response y[n-k] to a shifted version of the input x[n-k] is identical to a shifted

version of the response y[n] for the unshifted input x[n].

Time invariant: y[n-k] = y[n-k]

Time variant: y[n-k] y[n-k]

Linear & Nonlinear:

Linear:

Should satisfy superposition principle.

The response of the system to a weighted sum of signals should be equal to the

corresponding weighted sum of the outputs to each of the individual input signals.

H{a1x1[n] + a2x2[n]} = a1H{x1[n]} + a2H{x2[n]}

Nonlinear: It does not satisfy the superposition principle.

Delay H

x[n]x[n-k] y[n-k]

H Delay

x[n]y[n] y[n-k]

x1[n]

x2[n]

x1[n]

x2[n]

a1

a2

a1

a2

+

+

H

H

H

y[n]

y[n]

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Causal & Non-causal:

Causal:

The output at any time depends only on present and past inputs, but does not

depend on future inputs.

Ex: y[n] = 7x[n] 9x[n-3]

Non-causal:

The output at any time depends not only on present and past inputs but also on

future inputs unrealistic.

Ex: y[n] = 3x[n] + 6x[n+3]

Stable & Unstable:

Stable:

A system is said to be Bounded Input Bounded Output (BIBO) stable if and only

if every bounded input produces a bounded output

Bounded means finite

System is stable if the impulse response is absolutely summable. (i.e.)

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TheZTransform

The bilateral z transform or two-sided z transform of the discrete-time signal x[n]

is defined to be

where is a complex variable. Since signals are typically defined to begin (become

nonzero) at time , and since filters are often assumed to be causal, the lower

summation limit given above may be written as 0 rather than - to yield the unilateral z

transform or one-sided z transform:

Theztransform of a signal can be regarded as apolynomialin , with coefficients

given by the signal samples. For example, the finite-duration signal

has the z-transform,

.

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Z-transform:

Here z is the complex variable. The z-transform of x (n) is also represented by operator Z

i.e.,

X (z) and x (n) is called a z-transform pair and it is represented as follows

This z-transform is also called as bilateral or two sided z-transform.

Region of Convergence:

The values of z for which the z-transform converges is called region of

convergence (ROC).

The z-transform is an infinite power series.

This series is not convergent for all values of z always.

Hence it is necessary to mention the region of convergence (ROC) along with z-

transform.

(a) Right-sided (causal) (b) Left-sided (anti-causal) (c) Two-sided-(non-causal)

Importance of ROC Specification

1. The importance of the ROC cannot be overemphasized. It is partof the Z-transform.

2. In specifying the Z-transform x( z) of a signal x( n), the ROC must be given -

otherwise, the Z-transform cannot be inverted- in order to re-obtain x( n) from

X(z), the ROC must be given.

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Example - Consider two sequences

Here it is important to understand that X(z) Y(z). The ROCs of these two

transforms do not even intersect. Not equal! X( z) + ROC unique x( n).

Other Properties of ROC

ROC Shape is either Ring or disk in the z-plane centered at the origin.

Fourier Transform converges absolutely if and only if the ROC of the Z-transform of

x(n) includes unite circle.

ROC cannot contain poles.

ROC Region must be a connected region.

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Properties of z-transform:

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Properties of ROC

Whether the z-transformX(z) of a function x[n] exists depends on whether or not

the transform summation converges

which in turn depends on the duration and magnitude ofx[n] as well as the magnitude

. Ifx[n] is right sided (i.e.,x[n]=0 forn0, and the larger |z| the more quickly x[n]z-n decays as . On the other

hand, ifx[n] is left sided (i.e.,x[n]=0 forn>n0), it may have infinite duration fornr0 is also in the ROC.

Ifx[n] is left sided and the circle |z|=r0 is in the ROC, then any zfor which 0

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Ifx[n] is two-sided, then the ROC is the intersection of the two one-sided ROCs

corresponding to the two one-sided parts ofx[n]. This intersection can be either a

ring or an empty set.

IfX(z) is rational, then its ROC does not contain any poles (by definition

dose not exist). The ROC is bounded by the poles or extends

to infinity.

IfX(z) is a rational z-transform of a right sided function x[n], then the ROC is the

region outside the outmost pole. Ifx[n]=0 forn0, when , forn

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=

=

The ROC for both parts combined is the intersection of the individual ROCs:

|b|