Digital Signal Processing - Quick Guide

Definition

Anything that carries information can be called as signal. It can also be defined as a physical quantity that varies with time, temperature, pressure or with any independent variables such as speech signal or video signal.

The process of operation in which the characteristics of a signal (Amplitude, shape, phase, frequency, etc.) undergoes a change is known as signal processing.

Note − Any unwanted signal interfering with the main signal is termed as noise. So, noise is also a signal but unwanted.

According to their representation and processing, signals can be classified into various categories details of which are discussed below.

Continuous Time Signals

Continuous-time signals are defined along a continuum of time and are thus, represented by a continuous independent variable. Continuous-time signals are often referred to as analog signals.

This type of signal shows continuity both in amplitude and time. These will have values at each instant of time. Sine and cosine functions are the best example of Continuous time signal.

The signal shown above is an example of continuous time signal because we can get value of signal at each instant of time.

Discrete Time signals

The signals, which are defined at discrete times are known as discrete signals. Therefore, every independent variable has distinct value. Thus, they are represented as sequence of numbers.

Although speech and video signals have the privilege to be represented in both continuous and discrete time format; under certain circumstances, they are identical. Amplitudes also show discrete characteristics. Perfect example of this is a digital signal; whose amplitude and time both are discrete.

The figure above depicts a discrete signal’s discrete amplitude characteristic over a period of time. Mathematically, these types of signals can be formularized as;

$$x = \left \{ x\left [ n \right ] \right \},\quad -\infty < n< \infty$$

Where, n is an integer.

It is a sequence of numbers x, where n^th number in the sequence is represented as x[n].

Digital Signal Processing - Basic CT Signals

To test a system, generally, standard or basic signals are used. These signals are the basic building blocks for many complex signals. Hence, they play a very important role in the study of signals and systems.

Unit Impulse or Delta Function

A signal, which satisfies the condition, $\delta(t) = \lim_{\epsilon \to \infty} x(t)$ is known as unit impulse signal. This signal tends to infinity when t = 0 and tends to zero when t ≠ 0 such that the area under its curve is always equals to one. The delta function has zero amplitude everywhere excunit_impulse.jpgept at t = 0.

Properties of Unit Impulse Signal

δ(t) is an even signal.
δ(t) is an example of neither energy nor power (NENP) signal.
Area of unit impulse signal can be written as;
Weight or strength of the signal can be written as;
Area of the weighted impulse signal can be written as −

Unit Step Signal

A signal, which satisfies the following two conditions −

$U(t) = 1(when\quad t \geq 0 )and$
$U(t) = 0 (when\quad t < 0 )$

is known as a unit step signal.

It has the property of showing discontinuity at t = 0. At the point of discontinuity, the signal value is given by the average of signal value. This signal has been taken just before and after the point of discontinuity (according to Gibb’s Phenomena).

If we add a step signal to another step signal that is time scaled, then the result will be unity. It is a power type signal and the value of power is 0.5. The RMS (Root mean square) value is 0.707 and its average value is also 0.5

Ramp Signal

Integration of step signal results in a Ramp signal. It is represented by r(t). Ramp signal also satisfies the condition $r(t) = \int_{-\infty}^{t} U(t)dt = tU(t)$. It is neither energy nor power (NENP) type signal.

Parabolic Signal

Integration of Ramp signal leads to parabolic signal. It is represented by p(t). Parabolic signal also satisfies he condition $p(t) = \int_{-\infty}^{t} r(t)dt = (t^{2}/2)U(t)$ . It is neither energy nor Power (NENP) type signal.

Signum Function

This function is represented as

$$sgn(t) = \begin{cases}1 & for\quad t >0\\-1 & for\quad t<0\end{cases}$$

It is a power type signal. Its power value and RMS (Root mean square) values, both are 1. Average value of signum function is zero.

Sinc Function

It is also a function of sine and is written as −

$$SinC(t) = \frac{Sin\Pi t}{\Pi T} = Sa(\Pi t)$$

Properties of Sinc function

It is an energy type signal.
$Sinc(0) = \lim_{t \to 0}\frac{\sin \Pi t}{\Pi t} = 1$
$Sinc(\infty) = \lim_{t \to \infty}\frac{\sin \Pi \infty}{\Pi \infty} = 0$ (Range of sinπ∞ varies between -1 to +1 but anything divided by infinity is equal to zero)
If $ \sin c(t) = 0 => \sin \Pi t = 0$

$\Rightarrow \Pi t = n\Pi$

$\Rightarrow t = n (n \neq 0)$

Sinusoidal Signal

A signal, which is continuous in nature is known as continuous signal. General format of a sinusoidal signal is

$$x(t) = A\sin (\omega t + \phi )$$

Here,

A = amplitude of the signal

ω = Angular frequency of the signal (Measured in radians)

φ = Phase angle of the signal (Measured in radians)

The tendency of this signal is to repeat itself after certain period of time, thus is called periodic signal. The time period of signal is given as;

$$T = \frac{2\pi }{\omega }$$

The diagrammatic view of sinusoidal signal is shown below.

Rectangular Function

A signal is said to be rectangular function type if it satisfies the following condition −

$$\pi(\frac{t}{\tau}) = \begin{cases}1, & for\quad t\leq \frac{\tau}{2}\\0, & Otherwise\end{cases}$$ Rectangular Function

Being symmetrical about Y-axis, this signal is termed as even signal.

Triangular Pulse Signal

Any signal, which satisfies the following condition, is known as triangular signal.

$$\Delta(\frac{t}{\tau}) = \begin{cases}1-(\frac{2|t|}{\tau}) & for|t|<\frac{\tau}{2}\\0 & for|t|>\frac{\tau}{2}\end{cases}$$ Triangular Pulse Signal

This signal is symmetrical about Y-axis. Hence, it is also termed as even signal.

Digital Signal Processing - Basic DT Signals

We have seen that how the basic signals can be represented in Continuous time domain. Let us see how the basic signals can be represented in Discrete Time Domain.

Unit Impulse Sequence

It is denoted as δ(n) in discrete time domain and can be defined as;

$$\delta(n)=\begin{cases}1, & for \quad n=0\\0, & Otherwise\end{cases}$$ Unit Impulse Sequence

Unit Step Signal

Discrete time unit step signal is defined as;

$$U(n)=\begin{cases}1, & for \quad n\geq0\\0, & for \quad n<0\end{cases}$$ Unit Step Signal

The figure above shows the graphical representation of a discrete step function.

Unit Ramp Function

A discrete unit ramp function can be defined as −

$$r(n)=\begin{cases}n, & for \quad n\geq0\\0, & for \quad n<0\end{cases}$$ Unit Ramp Function

The figure given above shows the graphical representation of a discrete ramp signal.

Parabolic Function

Discrete unit parabolic function is denoted as p(n) and can be defined as;

$$p(n) = \begin{cases}\frac{n^{2}}{2} ,& for \quad n\geq0\\0, & for \quad n<0\end{cases}$$

In terms of unit step function it can be written as;

$$P(n) = \frac{n^{2}}{2}U(n)$$ Parabolic Function

The figure given above shows the graphical representation of a parabolic sequence.

Sinusoidal Signal

All continuous-time signals are periodic. The discrete-time sinusoidal sequences may or may not be periodic. They depend on the value of ω. For a discrete time signal to be periodic, the angular frequency ω must be a rational multiple of 2π.

A discrete sinusoidal signal is shown in the figure above.

Discrete form of a sinusoidal signal can be represented in the format −

$$x(n) = A\sin(\omega n + \phi)$$

Here A,ω and φ have their usual meaning and n is the integer. Time period of the discrete sinusoidal signal is given by −

$$N =\frac{2\pi m}{\omega}$$

Where, N and m are integers.

DSP - Classification of CT Signals

Continuous time signals can be classified according to different conditions or operations performed on the signals.

Even and Odd Signals

Even Signal

A signal is said to be even if it satisfies the following condition;

$$x(-t) = x(t)$$

Time reversal of the signal does not imply any change on amplitude here. For example, consider the triangular wave shown below.

The triangular signal is an even signal. Since, it is symmetrical about Y-axis. We can say it is mirror image about Y-axis.

Consider another signal as shown in the figure below.

We can see that the above signal is even as it is symmetrical about Y-axis.

Odd Signal

A signal is said to be odd, if it satisfies the following condition

$$x(-t) = -x(t)$$

Here, both the time reversal and amplitude change takes place simultaneously.

In the figure above, we can see a step signal x(t). To test whether it is an odd signal or not, first we do the time reversal i.e. x(-t) and the result is as shown in the figure. Then we reverse the amplitude of the resultant signal i.e. –x(-t) and we get the result as shown in figure.

If we compare the first and the third waveform, we can see that they are same, i.e. x(t)= -x(-t), which satisfies our criteria. Therefore, the above signal is an Odd signal.

Some important results related to even and odd signals are given below.

Even × Even = Even
Odd × Odd = Even
Even × Odd = Odd
Even ± Even = Even
Odd ± Odd = Odd
Even ± Odd = Neither even nor odd

Representation of any signal into even or odd form

Some signals cannot be directly classified into even or odd type. These are represented as a combination of both even and odd signal.

$$x(t)\rightarrow x_{e}(t)+x_{0}(t)$$

Where x_e(t) represents the even signal and x_o(t) represents the odd signal

$$x_{e}(t)=\frac{[x(t)+x(-t)]}{2}$$

And

$$x_{0}(t)=\frac{[x(t)-x(-t)]}{2}$$

Example

Find the even and odd parts of the signal $x(n) = t+t^{2}+t^{3}$

Solution − From reversing x(n), we get

$$x(-n) = -t+t^{2}-t^{3}$$

Now, according to formula, the even part

$$x_{e}(t) = \frac{x(t)+x(-t)}{2}$$

$$= \frac{[(t+t^{2}+t^{3})+(-t+t^{2}-t^{3})]}{2}$$

$$= t^{2}$$

Similarly, according to formula the odd part is

$$x_{0}(t)=\frac{[x(t)-x(-t)]}{2}$$

$$= \frac{[(t+t^{2}+t^{3})-(-t+t^{2}-t^{3})]}{2}$$

$$= t+t^{3}$$

Periodic and Non-Periodic Signals

Periodic Signals

Periodic signal repeats itself after certain interval of time. We can show this in equation form as −

$$x(t) = x(t)\pm nT$$

Where, n = an integer (1,2,3……)

T = Fundamental time period (FTP) ≠ 0 and ≠∞

Fundamental time period (FTP) is the smallest positive and fixed value of time for which signal is periodic.

A triangular signal is shown in the figure above of amplitude A. Here, the signal is repeating after every 1 sec. Therefore, we can say that the signal is periodic and its FTP is 1 sec.

Non-Periodic Signal

Simply, we can say, the signals, which are not periodic are non-periodic in nature. As obvious, these signals will not repeat themselves after any interval time.

Non-periodic signals do not follow a certain format; therefore, no particular mathematical equation can describe them.

Energy and Power Signals

A signal is said to be an Energy signal, if and only if, the total energy contained is finite and nonzero (0<E<∞). Therefore, for any energy type signal, the total normalized signal is finite and non-zero.

A sinusoidal AC current signal is a perfect example of Energy type signal because it is in positive half cycle in one case and then is negative in the next half cycle. Therefore, its average power becomes zero.

A lossless capacitor is also a perfect example of Energy type signal because when it is connected to a source it charges up to its optimum level and when the source is removed, it dissipates that equal amount of energy through a load and makes its average power to zero.

For any finite signal x(t) the energy can be symbolized as E and is written as;

$$E = \int_{-\infty}^{+\infty} x^{2}(t)dt$$

Spectral density of energy type signals gives the amount of energy distributed at various frequency levels.

Power type Signals

A signal is said to be power type signal, if and only if, normalized average power is finite and non-zero i.e. (0<p<∞). For power type signal, normalized average power is finite and non-zero. Almost all the periodic signals are power signals and their average power is finite and non-zero.

In mathematical form, the power of a signal x(t) can be written as;

$$P = \lim_{T \rightarrow \infty}1/T\int_{-T/2}^{+T/2} x^{2}(t)dt$$

Difference between Energy and Power Signals

The following table summarizes the differences of Energy and Power Signals.

Power signal	Energy Signal
Practical periodic signals are power signals.	Non-periodic signals are energy signals.
Here, Normalized average power is finite and non-zero.	Here, total normalized energy is finite and non-zero.
Mathematically, $$P = \lim_{T \rightarrow \infty}1/T\int_{-T/2}^{+T/2} x^{2}(t)dt$$	Mathematically, $$E = \int_{-\infty}^{+\infty} x^{2}(t)dt$$
Existence of these signals is infinite over time.	These signals exist for limited period of time.
Energy of power signal is infinite over infinite time.	Power of the energy signal is zero over infinite time.

Solved Examples

Example 1 − Find the Power of a signal $z(t) = 2\cos(3\Pi t+30^{o})+4\sin(3\Pi +30^{o})$

Solution − The above two signals are orthogonal to each other because their frequency terms are identical to each other also they have same phase difference. So, total power will be the summation of individual powers.

Let $z(t) = x(t)+y(t)$

Where $x(t) = 2\cos (3\Pi t+30^{o})$ and $y(t) = 4\sin(3\Pi +30^{o})$

Power of $x(t) = \frac{2^{2}}{2} = 2$

Power of $y(t) = \frac{4^{2}}{2} = 8$

Therefore, $P(z) = p(x)+p(y) = 2+8 = 10$…Ans.

Example 2 − Test whether the signal given $x(t) = t^{2}+j\sin t$ is conjugate or not?

Solution − Here, the real part being t² is even and odd part (imaginary) being $\sin t$ is odd. So the above signal is Conjugate signal.

Example 3 − Verify whether $X(t)= \sin \omega t$ is an odd signal or an even signal.

Solution − Given $X(t) = \sin \omega t$

By time reversal, we will get $\sin (-\omega t)$

But we know that $\sin(-\phi) = -\sin \phi$.

Therefore,

$$\sin (-\omega t) = -\sin \omega t$$

This is satisfying the condition for a signal to be odd. Therefore, $\sin \omega t$ is an odd signal.

DSP - Classification of DT Signals

Just like Continuous time signals, Discrete time signals can be classified according to the conditions or operations on the signals.