 Digital Signal Processing Tutorial
 DSP  Home
 DSP  SignalsDefinition
 DSP  Basic CT Signals
 DSP  Basic DT Signals
 DSP  Classification of CT Signals
 DSP  Classification of DT Signals
 DSP  Miscellaneous Signals
 Operations on Signals
 Operations Signals  Shifting
 Operations Signals  Scaling
 Operations Signals  Reversal
 Operations Signals  Differentiation
 Operations Signals  Integration
 Operations Signals  Convolution
 Basic System Properties
 DSP  Static Systems
 DSP  Dynamic Systems
 DSP  Causal Systems
 DSP  NonCausal Systems
 DSP  AntiCausal Systems
 DSP  Linear Systems
 DSP  NonLinear Systems
 DSP  TimeInvariant Systems
 DSP  TimeVariant Systems
 DSP  Stable Systems
 DSP  Unstable Systems
 DSP  Solved Examples
 ZTransform
 ZTransform  Introduction
 ZTransform  Properties
 ZTransform  Existence
 ZTransform  Inverse
 ZTransform  Solved Examples
 Discrete Fourier Transform
 DFT  Introduction
 DFT  Time Frequency Transform
 DTF  Circular Convolution
 DFT  Linear Filtering
 DFT  Sectional Convolution
 DFT  Discrete Cosine Transform
 DFT  Solved Examples
 Fast Fourier Transform
 DSP  Fast Fourier Transform
 DSP  InPlace Computation
 DSP  Computer Aided Design
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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; $$A = \int_{\infty}^{\infty} \delta (t)dt = \int_{\infty}^{\infty} \lim_{\epsilon \to 0} x(t) dt = \lim_{\epsilon \to 0} \int_{\infty}^{\infty} [x(t)dt] = 1$$
 Weight or strength of the signal can be written as; $$y(t) = A\delta (t)$$
 Area of the weighted impulse signal can be written as − $$y (t) = \int_{\infty}^{\infty} y (t)dt = \int_{\infty}^{\infty} A\delta (t) = A[\int_{\infty}^{\infty} \delta (t)dt ] = A = 1 = Wigthedimpulse$$
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 −
$$\pi(\frac{t}{\tau}) = \begin{cases}1, & for\quad t\leq \frac{\tau}{2}\\0, & Otherwise\end{cases}$$Being symmetrical about Yaxis, this signal is termed as even signal.
Triangular Pulse Signal
Any signal, which satisfies the following condition, is known as triangular signal.
$$\Delta(\frac{t}{\tau}) = \begin{cases}1(\frac{2t}{\tau}) & fort<\frac{\tau}{2}\\0 & fort>\frac{\tau}{2}\end{cases}$$This signal is symmetrical about Yaxis. Hence, it is also termed as even signal.