## Basic operations on continuous time signal

In this section we will study some basic operations on continuous time signal:

### A)     Operation performed on dependent variables:

These operations include sum, product, difference, even, odd, etc.

#### 1)      Amplitude scaling:

Amplitude scaling means changing an amplitude of given continuous time signal. We will denote continuous time signal by x(t). If it is multiplied by some constant ‘B’ then resulting signal is,

y(t)= B x(t)

Example: Sketch y(t) = 5u(t)

Solution: we know that u(t) is unit step function. So if we multiply it with 5, its amplitude will become 5 and it shown as follows:

amplitude scaling

#### 2)      Sum and difference of two signals:

Consider two signals x1(t) and x2(t). Then addition of these signals is denoted by y(t)=x1(t)+x2(t). similarly subtraction is given by y(t)=x1(t)-x2(t).

Example: Sketch y(t) = u(t) – u(t – 2)

Solution: First, plot each of the portions of this signal separately

• x1(t) = u(t) …….Simply a step signal

• x2(t) = –u(t-2) ……. Delayed step signal by 2 units and multiplied by -1.

Then, move from one side to the other, and add their instantaneous values:

Sum and difference of two signals

#### 3)      Product of two signals:

If x1(t) and x2(t) are two continuous signals then the product of x1(t) and x2(t) is,

Y(t) = x1(t)x2(t).

Example:  Sketch y(t) = u(t)·u(t – 2)

Solution: First, plot each of the portions of this signal separately

• x1(t) = u(t) _ Simply a step signal

• x2(t) = u(t-2) _ Delayed step signal

Then, move from one side to the other, and multiply instantaneous values:

Multiplication of two signals

#### 4)      Even and odd parts:

Even part of signal x(t) is given by,

And odd part of x(t) is given by,

### B)     Operations performed on independent variables:

#### 1)      Time shifting:

A signal x(t) is said to be ‘shifted in time’ if we replace t by (t-T). thus x(t-T) represents the time shifted version of x(t) and the amount of time shift is ‘T’ sec. if T is positive then the shift is to right (delay) and if T is negative then the shift is to the left (advance).

Example:  Sketch y(t) = u(t – 2)

solution:

#### 2)      Time scaling:

The compression or expression of a signal in time is known as the time scaling. If x(t is the original signal then x(at) represents its time scaled version. Where a is constant.

If a> 1 then x(at) will be a compressed version of x(t) and if a< 1 then it will be a expanded version of x(t).

Example:  Let x(t) = u(t) – u(t – 2). Sketch y(t) = x(t/2)

solution:

3)      Time reversal (Time inversion):

Flips the signal about the y axis. y(t) = x(-t) .

Example: Let x(t) = u(t), and perform time reversal.

solution:

## Basic signals

In the analysis of communication system, standard test signals play very important role. Such signals are used to check the performance of the system. Applying such signals at the system; the output is checked. Now depending on the input-output characteristic of that particular system; study of different properties of a system can be done. Some standard test signals are as follows:

• Delta or unit impulse function
• Unit step signal
• Unit ramp signal
• Exponential signal
• Sinusoidal signal

## Delta or unit impulse function:

• A discrete time unit impulse function is denoted by d(n). Its amplitude is 1 at n=0 and for all other values of n; its amplitude is zero.

d(n)= 1 for n=0

d(n)=0 for n?0

In the sequence form it can be represented as,

d(n) = {….,0,0,0,1,0,0,0….}
or d(n) = {1}

The graphical representation of delta function for D.T. signal  is as shown in figure below:

• A continuous time delta function is denoted by d(t). mathematically it is expressed as follows:

d(t)=1 for t=0

d(t)=0 for t?0

The graphical representation of delta function for C.T. signal is shown in figure below:

## Unit step signal:

Unit step signal means the signal has unit amplitude for positive axis and has zero amplitude for negative axis.

There are two types of unit step signal as follows:

1) Discrete time unit step signal

2) Continuous time unit step signal

Let us discuss these two types one by one:

## 1) Discrete time unit step signal

• A discrete time unit step signal is denoted by u(n). its value is unity (1) for all positive values of n. that means its value is one for n = 0. While for other values of n, its value is zero.

u(n)= 1 for  n = 0

u(n)= 0 for n < 0

In the form of sequence it can written as,

u(n) = {1,1,1,1,….}

Graphically it can be represented as follows:

## 2) Continuous time unit step signal

• A continuous type unit step signal is denoted by u(t). mathematically it can be expressed as,

u(t)= 1 for t = 0

u(t) = 0 for t < 0

it is shown in figure below:

## Unit ramp signal:

• A discrete time unit ramp signal is denoted by ur(n). its value increases linearly with sample number n. mathematically it is defined as,

Ur(n)= n for n = 0

Ur(n) = 0 for n < 0

From above equation, it is clear that the value of signal at a particular interval is equal to the number of interval at that instant. for example; for first interval signal has amplitude 1, for second it has amplitude 2, for third it is 3, and so on.

Graphically it is represented in figure below:

• A continuous time ramp type signal is denoted by r(t). mathematically it is expressed as,

r(t) = 1 for t = 0

r(t) = 0 for t < 0

From above equation, it is clear that the value of signal at a particular time is equal to the time at that instant. for example; for one second signal has amplitude 1, for two second it has amplitude 2, for third it is 3, and so on.

It is shown in figure below:

## Exponential signal:

In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative.

A discrete time exponential signal is expressed as,

Here ‘a’ is some real constant.

If ‘a’ is the complex number then x(n) is written as,

Here ? denotes the phase. Now depending upon value of ? we have different cases:

case 1) : when a>1

a>1

Since the signal is exponentially growing; it is called rising exponential signal.

case 2) : when 0<a<1

Since the signal is exponentially decreases; it is called decaying exponential signal.

case 3) : when a<-1

As shown in figure above, both signals are growing hence it is called as double sided growing exponential signal.

case 4) : when -1<a<0

As shown in figure above, both signals are decreases hence it is called as double sided decaying exponential signal.

• If we talk about continuous time exponential signals; everything is same as that of discrete time signals except it has continuous amplitude.

## Sinusoidal waveform:

A sinusoidal signal has the same shape as the graph of the sine function used in trigonometry. Sinusoidal signals are produced by rotating electrical machines such as dynamos and power station turbines and electrical energy is transmitted to the consumer in this form. In electronics, sine waves are among the most useful of all signals in testing circuits and analyzing system performance.

There are two types of sinusoidal waveforms:

1) Discrete time sinusoidal waveforms, and

2) Continuous time sinusoidal waveforms.

Let us discuss these waveforms one by one:

• Discrete time sinusoidal wave:
• Signals typically represent physical variables such as displacement, velocity, pressure, energy.
•  In most cases, we are concerned with variables that are time-dependent.
•  The discrete-time or digital time index is generally specified by tn = n T, where n is an integer and T is the sampling interval or period.
•  It is common to drop the explicit reference to T (or assume T = 1) and index discrete-time signals by the letter n.
• A discrete-time signal which is only dependent on time can be represented by x[n] for n = 0, 1, 2, …

A discrete time sinusoidal waveform is denoted by,

x(n) = A sin (wn)

Here a = amplitude and w = angular frequency = 2pf

This waveform is as shown in figure below:

Similarly a discrete time cosine waveform is expressed as,

x(n) = A cos (wn)

• Continuous time sinusoidal signals:

The sinusoidal signals include sine and cosine signals. They are as shown in figure below:

Mathematically they can be expressed as follows:

A sine signal: x(t) = A sin (wt) = A sin (2pft)

A cosine signal: x(t) = A cos (wt) = A cos (2pft)

## Definition of Signal

In a communication system, the word ‘signal’ is commonly used. Therefore we must know its exact meaning.

• Mathematically, signal is described as a function of one or more independent variables.
• Basically it is a physical quantity. It varies with some dependent or independent variables.
• So the term signal is defined as “A physical quantity which contains some information and which is function of one or more independent variables.”

## Classification of Signals:

There are various types of signals. Every signal has its own characteristics. The processing of signals mainly depends on the characteristic of that particular signal. So classification of signal is necessary. Broadly the signals are classified as below:

• Continuous and discrete time signals
• Continuous valued and discrete valued signals
• Periodic and non-periodic signals
• Even and odd signals
• Energy and power signals
• Deterministic and random signals
• Multichannel and multidimensional signals

## Continuous and discrete time signals:

### Continuous signal:

A signal of continuous amplitude is called continuous signal or analog signal. Continuous signal has some value at every instant of time.

### Examples:

Sine wave, cosine wave, triangular wave etc. similarly some electrical signals derived from physical quantities like temperature, pressure, sound etc. are also an examples of continuous signals.

### Mathematical expression:

Mathematically a continuous signal can be expressed as,

x(t)=A sin(wt+?)

Here A= amplitude of signal

w = angular frequency=2pf

?= phase shift

### Characteristics:

• For every fix value of t, x(t) is periodic in nature.
• If the frequency (1/t) is increased then the rate of oscillation also changes.

### Discrete time signal:

In this case the value of signal is specified only at specific time. So signal represented at “discrete interval of time” is called as discrete time of signal.

The discrete time signal is generated from continuous time signal by using the sampling operation. This process is shown in figure below.

• Consider a continuous analog signal as shown in figure a). This signal is continuous in nature from –infinity to +infinity.
• The sampling pulses are shown in figure b). These are train of pulses. Here the samples are taken with Ts as sampling time.
• Figure c) shows the discrete time signal
• For signal shown in figure a), the expression is x(t)=A cos (wt)
• And for signal shown in fig c) , the expression is x(t)= A cos (wn)

### Characteristics:

• Discrete time sinusoidal signals are identical when their frequencies are separated by integer multiple of 2p.
• If the frequency of discrete time sinusoidal is rational number, then such signal is periodic in nature.
• For the discrete time sinusoidal, the highest oscillation is obtained when angular frequency w=+p or –p.

## Continuous valued or discrete valued signals:

### Continuous valued signals:

If the variation in the amplitude of signal is continuous then, it is called continuous valued signal. Such signals may be continuous or discrete in nature. Following figure shows the examples of continuous valued signals.

### Discrete valued signals:

If the variation in the amplitude of signal is not continuous but the signal has certain discrete amplitude levels then such signal is called as discrete valued signal. Such signal may be again continuous or discrete in nature as shown in figure below.

## Periodic and Non-periodic signals:

### Periodic signal:

A signal which repeats itself after a fixed time period or interval is called as periodic signal. The periodicity of continuous time signal can be defined mathematically as,

x(t)=x(t+T0)

This is called as condition of periodicity. Here T0 is called as fundamental period. That means after this period signal repeats itself.

For the discrete time signal, the condition of periodicity is,

x(n)=x(n+N)

Here number ‘N’ is the period of signal. The smallest value of N for which the condition of periodicity exists is called fundamental period.

Following figure shows the examples of periodic signals:

### Non-periodic signals:

A signal which does not repeat itself after a fixed time period or does not repeat at all is called as non-periodic or aperiodic signal.

In other words we can say that, the period of non-periodic signal is infinity.

Following figure shows the non-periodic signal:

## Even and Odd Signals:

### Even signals:

An even signal is also called as symmetrical signal. A continuous time signal x(t) is said to be even or symmetrical if it satisfies the following condition:

Condition for symmetry: x(t)=x(-t)……..for continuous time signal.

Here x(-t) indicates that the signal is present for negative time period. That means x(-t) is the signal which is reflected about vertical axis.

Condition for symmetry: x(n)=x(-n)…….for discrete time signal.

Following figure shows the even signal:

### Odd signal:

A continuous time signal x(t) is said to be odd signal if it satisfies following condition:

Condition for odd signal: x(-t)=-x(t)…….for continuous time signal.

Here x(-t) indicates that the signal is present for negative time period. While –x(t) indicates that the amplitude of the signal is negative. Thus odd signal is not symmetric about vertical axis.

Condition for odd signal: x(-n)=-x(n)……for discrete time signal.

Following figure shows the odd signal:

Note: amplitude of odd signal at origin is always zero.