System in Continuous Time-domain

A system is defined as an entity that acts on an input signal x(t) and Transforms it into an output signal y(t)

In a system, there exists Cause and effect relationship between two (or) more signals

ex:- Electrical systems, mechanical systems etc.

A system is represented by a block, which accepts signals as inputs and produces signals as outputs.

input signal:- is called as excitation , source 9or) driving function -x(t)

output signal:- is called as response-y(t)

systems may be single input single output systems (or) Multi input Multi output systems.

Here we discuss only about SISO systems only.

y(t)=T(x(t)).

T- Transformation of x(t) (Or) operation on x(t).

In general systems are either Continuous -Time (or) Discrete Time systems.

continuous-Time systems (CT system) Discrete-Time systems (DT system)
A system that operates/produces on Continuous Time signals is a CT system. A system that operates/produces on Discrete Time signals  and produces a DT signals is a  DT system.

Classification of systems:-

Systems are classified into

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Fourier Series and it’s applications

The starting point of Fourier Series is the development of representation of signals as linear combination (sum of) of a set of basic signals.

f(t)\approx C_{1}x_{1}(t)+C_{2}x_{2}(t)+.......+C_{n}x_{n}(t)+....

The alternative representation if  a set of complex exponentials are used,

f(t)\approx C_{1}e^{j\omega _{o}t}+C_{2}e^{2j\omega _{o}t}+.......+C_{n}e^{jn\omega _{o}t}+....

The resulting representations are known as Fourier Series in Continuous-Time [Fourier Transform in the case of Non-Periodic signal]. Here we focus on representation of Continuous-Time and Discrete-Time periodic signals in terms of basic signals as Fourier Series and extend the analysis to the Fourier Transform representation of broad classes of aperiodic, finite energy signals.

These Fourier Series & Fourier Transform representations are most powerful tools used

  1. In the analyzation of signals and LTI systems.
  2. Designing of Signals & Systems.
  3. Gives insight to S&S.

The development of Fourier series analysis has a long history involving a great many individuals and the investigation of many different physical phenomena.

The concept of using “Trigonometric Sums”, that is sum of harmonically related sines and cosines (or) periodic complex exponentials are used to predict astronomical events.

Similarly, if we consider the vertical deflection f(t,x) of the string at time t and at a distance x along the string then for any fixed instant of time, the normal modes are harmonically related sinusoidal functions of x.

The scientist Fourier’s work, which motivated him physically was the phenomenon of heat propagation and diffusion. So he found that the temperature distribution through a body can be represented by using harmonically related sinusoidal signals.

In addition to this he said that any periodic signal could be represented by such a series.

Fourier obtained a representation for aperiodic (or) non-periodic signals not as weighted sum of harmonically related sinusoidals but as weighted integrals of Sinusoids that are not harmonically related, which is known as Fourier Integral (or) Fourier Transform.

In mathematics, we use the analysis of Fourier Series and Integrals in 

  1. The theory of Integration.
  2. Point-set topology.
  3. and in the eigen function expansions.

In addition to the original studies of vibration and heat diffusion, there are numerous other problems in science and Engineering in which sinusoidal signals arise naturally, and therefore Fourier Series and Fourier T/F’s plays an important role.

for example, Sine signals arise naturally in describing the motion of the planets and the periodic behavior of the earth’s climate.

A.C current sources generate sinusoidal signals as voltages and currents. As we will see the tools of Fourier analysis enable us to analyze the response of an LTI system such as a circuit to such Sine inputs.

Waves in the ocean consists of the linear combination of sine waves with different spatial periods (or) wave lengths.

Signals transmitted by radio and T.V stations are sinusoidal in nature as well.

The problems of mathematical physics focus on phenomena in Continuous Time, the tools of Fourier analysis for DT signals and systems have their own distinct historical roots and equally rich set of applications.

In particular, DT concepts and methods are fundamental to the discipline of numerical analysis , formulas for the processing of discrete sets of data points to produce numerical approximations for interpolation and differentiation were being investigated.

FFT known as Fast Fourier Transform algorithm was developed, which suited perfectly for efficient digital implementation and it reduced the time required to compute transform by orders of magnitude (which utilizes the DTFS and DTFT practically).

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few problems on Auto correlatioon Function(ACF) and Energy Spectral Density(ESD)

  1. Find the Auto correlation function of x(t) = \frac{1}{\sqrt{2\pi }}\exp ^{\frac{-t^{2}}{2}}.

Ans. We know that  Auto correlation function forms fourier transform pair with Energy Spectral Density function

ACF\leftrightarrow ESD

R_{xx}(t)\leftrightarrow S(f)

the Fourier Transform of  e^{-ct^{2}}\leftrightarrow \frac{\sqrt{\pi }}{c}e^{-\pi ^{2}f^{2}}

\frac{1}{\sqrt{2\pi }}e^{\frac{-t^{2}}{2}}\leftrightarrow \frac{1}{\sqrt{2\pi }}.\sqrt{\frac{\pi }{(1/2)}}e^{\frac{-\pi ^{2}f^{2}}{(1/2)}} here c = \frac{1}{2}

\frac{1}{\sqrt{2\pi }}e^{\frac{-t^{2}}{2}}\leftrightarrow \frac{1}{\sqrt{2\pi }}.\sqrt{2\pi }e^{-2 \pi ^{2}f^{2}}

\frac{1}{\sqrt{2\pi }}e^{\frac{-t^{2}}{2}}\leftrightarrow e^{-2 \pi ^{2}f^{2}}

x(t)\leftrightarrow X(f)

\therefore the Fourier Transform of x(t) is X(f) and is X(f) = e^{-2\pi ^{2}f^{2}} and the Energy Spectral Density S(f) = \left | X(f) \right |^{2}

S(f) = e^{-4\pi ^{2}f^{2}}

By finding the inverse Fourier Transform of S(f) gives the Auto Correlation Function

S(f) = e^{\frac{-\pi ^{2}f^{2}}{(1/4)}}

e^{\frac{-\pi ^{2}f^{2}}{(1/4)}}\leftrightarrow \frac{e^{\frac{-t^{2}}{4}}}{\sqrt{4\pi }}

e^{\frac{-\pi ^{2}f^{2}}{(1/4)}}\leftrightarrow \frac{e^{\frac{-t^{2}}{4}}}{2\sqrt{\pi }}

\therefore the ACF of the given signal is inverse Fourier Transform of S(f) which is R_{xx}(t) = \frac{e^{\frac{-t^{2}}{4}}}{2\sqrt{\pi }}.

 

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Energy Spectal Density of rectangular pulse and solved problems

As we all know the Rectangular pulse is defined as x(t) = rect(\frac{t}{T}), exists for a duration of T sec symmetrical with respect to y-axis as shown 

Fourier Transform is X(f) 

X(f) = \int_{\frac{-T}{2}}^{\frac{T}{2}}x(t)e^{-j2\pi ft}dt

X(f) = \int_{\frac{-T}{2}}^{\frac{T}{2}} 1.e^{-j2\pi ft}dt

X(f) = \left [ \frac{e^{-j2\pi ft}}{j2\pi f} \right ] ^{\frac{T}{2}}_{\frac{-T}{2}}

X(f) = \frac{1}{\pi f}\left [ \frac{e^{j\pi ft}-e^{-j\pi ft}}{2j} \right ]

X(f) = \frac{T \sin \pi fT}{\pi fT}

X(f) = X(f) = sinc (\pi fT) 

\therefore The Energy Spectral Density of the given signal x(t) will be S(f) = \left | X(f) \right |^{2}

s(f)= T^{2}sinc^{2}(\pi fT).

Pb2. Let x(t) = sinc(T\pi t) and  y(t) = \frac{dx(t)}{dt} find the Energy Spectral Density S(f) of y(t).

Ans:-  The Energy Spectral Density of y(t)is S(f)

i.e,  S(f) = \left | Y(f) \right |^{2}

y(t) \leftrightarrow \frac{dx(t)}{dt}, then Fourier Tansform of  y(t)is

Y(f) \leftrightarrow j2\pi f X(f)

as x(t) = sinc(\pi Tt)\leftrightarrow X(f)= \frac{1}{T}rect(\frac{f}{T})

then Y(f) \leftrightarrow j2\pi f X(f)

\left | Y(f) \right |^{2} \leftrightarrow \left | j2\pi f \frac{1}{T}rect(\frac{f}{T}) \right | ^{2}

\left | Y(f) \right |^{2}\leftrightarrow \frac{4\pi ^{2}f^{2}}{T^{2}}rect^{2}(\frac{f}{T})

\therefore S(f) = \frac{4\pi ^{2}f^{2}}{T^{2}}rect^{2}(\frac{f}{T})

Pb3. The Energy contained with in the band [0, f], f>0 E(f) = \frac{1}{\sigma ^{3}}( {2-(2+2\sigma f+\sigma ^{2}f^{2})e^{-\sigma f}}) then find the ESD S(f), for any f>0 Energy of a signal can be defined \int_{-\infty }^{\infty }\left | S(f) \right | df

Ans:- Given E(f) = \int_{0}^{f}\left | S(f) \right | df

S(f ) = \frac{dE(f)}{df}

E(f) = \frac{1}{\sigma ^{3}}( {2-(2+2\sigma f+\sigma ^{2}f^{2})e^{-\sigma f}})

\frac{dE(f)}{df} = 0- \frac{1}{\sigma ^{3}}[2(-\sigma )e^{-\sigma f}+2\sigma f(-\sigma )e^{-\sigma f}+2\sigma e^{-\sigma f}+\sigma ^{2}f^{2}(-\sigma )e^{-\sigma f}+\sigma ^{2}(2f )e^{-\sigma f}]

\frac{dE(f)}{df} = - \frac{1}{\sigma ^{3}}[-2\sigma e^{-\sigma f}-2\sigma^{2} fe^{-\sigma f}+2\sigma e^{-\sigma f}+\sigma ^{3}f^{2}e^{-\sigma f}+2\sigma ^{2}fe^{-\sigma f}]

\frac{dE(f)}{df} = - \frac{1}{\sigma ^{3}}\sigma ^{3}f^{2}e^{-\sigma f}

\therefore S(f)=f^{2}e^{-\sigma f}

Pb:-find the Auto correlation function of  x(t) = \frac{1}{\sqrt{2\pi }}e^{\frac{-t^{2}}{2}}.

Ans:-  Auto Correlation Function \leftrightarrowEnergy spectral Density

R_{xx} (t) \leftrightarrow S(f)

we know that Fourier Transform of e^{-ct^{2}}\leftrightarrow \sqrt{\frac{\pi }{c}}e^{\frac{-\pi ^{2}f^{2}}{c}}

By using the above rule

\frac{1}{\sqrt{2\pi }}e^{\frac{-t^{2}}{2}}\leftrightarrow \frac{1}{\sqrt{2\pi }} \sqrt{\frac{\pi }{(1/2)}}e^{\frac{-\pi ^{2}f^{2}}{(1/2)}}, here  c=1/2

\frac{1}{\sqrt{2\pi }}e^{\frac{-t^{2}}{2}}\leftrightarrow \frac{1}{\sqrt{2\pi }} \sqrt{{2\pi }}e^{-2\pi ^{2}f^{2}}

\frac{1}{\sqrt{2\pi }}e^{\frac{-t^{2}}{2}}\leftrightarrow e^{-2\pi ^{2}f^{2}}

\therefore X(f) =e^{-2\pi ^{2}f^{2}}

Now the Energy Spectral Density S(f) = \left | X(f) \right |^{2}

S(f) =e^{-4\pi ^{2}f^{2}}

by finding the inverse Fourier Transform of S(f)

e^{-4\pi ^{2}f^{2}}\Rightarrow e^{\frac{-\pi ^{2}f^{2}}{(1/4)}}

e^{\frac{-\pi ^{2}f^{2}}{(1/4)}}\leftrightarrow \frac{1}{\sqrt{4\pi }}e^{\frac{-t^{2}}{4}}, c=1/4

Auto Correlation Function = \frac{1}{2\sqrt{\pi }}e^{\frac{-t^{2}}{4}}

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