Hello everyone, I hope that you all are doing good in your lives. The topic for today is the classification of signals and systems and also we are going to go through the basics of the elementary signal. We are gonna discuss both continuous as well as discrete signals. Before going into deep we must get clear about the simple terms.

Systems

A System is formally defined as an entity that processes one or more signals to get a function; therefore giving rise to a new signal. The interaction between the systems and the signals along it is shown in the figure below. The description of input and output signal naturally depends on the require application of the system For example;

Firstly In an automatic speaker recognition system, The input signal is a speech signal, the system is a computer and the output is the identity of the speaker.
However here, in this case, we are dealing with signals, we will have inputs and outputs both as signals; and the system as a black box that either contains a software/coder or a physical device that performs processing on the particular input function to gain desire output signal.

Note the input signal and output signal is express as excitation and response respectively. types of signals and systems

Types of System

Linear and Non – Linear systems (i.e. Typical RLC circuit)
Stable and Unstable system (i.e. Mass damper system)
Static and Dynamic System (i.e. Multiplexer and flip flops system)
And lastly Time variant and Time invariant System (i.e RC circuits)

Signals

A signal is a time-varying physical quantity which gives some information and is a function of one or more independent variable. A signal can be of any dimension (1D, 2D, 3D) and of any form.

For example:- Radio Signals, Current Signals, Voltage Signals, Microwave Signals, Exponential Signals.

Classification Of Signals

Continuous and Discrete Time Signal

Continuous Time signal [C.T]

A Continuous signal is a signal which is defined at every instant of time. In C.T the amplitude Vs time graph goes by the Waveform, and this waveform is continuous in nature.

Representation of sine wave in C.T form

x(t) = ASinω₀t
However here x is the signal;
() indicates continuous time;
and lastly t denotes an independent variable.

$x(t) = ASinwt = ASin (\frac {2*\pi} {\pi}) * t \end.$

$\therefore x(t) = ASint \end.$

Besides the exponential and sinusoidal waveform of C.T Signal is shown below; continuous signals and systems

C.T signals are used in analog circuits for example op-amp, integrator, differentiator, unity gain, etc.

Discrete Time Signal [D.T]

Discrete-time signals are signals or a set of signals which is defined at a particular interval of time. However, it is mostly obtained by sampling and thus it is only present at intervals of real value.

Thus the representation of D.T signal;

x[n] = {-1,0,1,2,3,4}
           ↑
Where x is signal;
[] indicates Discrete time;
and lastly n represents frequency for example (1,2,3,4,..)

Note:- the ↑ indicates value of x[n] at a zeroth instant. However, if it is not mentioned then by default consider the first sample as starting sample.

Lastly, the waveform below is D.T representations in sinusoidal and exponential form. Discrete signals and systems

D.T signals are use d in circuits like Microprocessor, Flip Flops, Shift Register, Counters, etc.

Sampling Theorem in signals and systems

A continuous-time signal x(t) can be represented in its sample form and can be reconstructed into its original form if and only if, “f_s” sampling frequency is ≥ 2W where W is the highest frequency component.

Minimum value of sampling rate = Nyquist Rate = 2W Hz

$Nyquist \quad interval = \frac {1} {Nyquist \quad Rate} \end.$

$\therefore \quad= \frac {1} {2W} secs\end.$

Effect of over sampling, Perfect Sampling and Under Sampling on signals and systems

Firstly we should go be knowing the spectrum of the original signal which is shown below; Spectrum of original signals

And now we can easily go through all other spectra’s i.e.; effects of sampling on signals

Aliasing in signals and systems

The phenomenon where high-frequency components interfere with the low-frequency components is called aliasing and certainly should be avoided.

However for avoiding aliasing;

f_s ≥ 2W
Lastly using anti-aliasing filter which is a low pass filter with a cutoff frequency of f_s/2

Thus the frequency of aliasing = f_s – W

Numerical on Nyquist Rate

Q.1 Find Nyquist rate and Nyquist interval for solving analog signals and systems;

x(t) = 4Cos(4000πt)

Solution:- Firstly comparing with;

$x(t) = A Cos \omega t \end.$

Therefore we get;

$w = 400 \pi \end.$

$\therefore 2 \pi f = 400 \pi \end.$

$\therefore f = 200Hz \end.$

$\therefore W = 2000Hz \end.$

$\therefore Nyquist \quad Rate = 2W = 4000Hz \end.$

$\therefore Nyquist \quad Interval = \frac {1} {4000} = 0.25 \quad millisec \end.$

However, from above, we conclude that the value of Nyquist rate and Nyquist interval is 4000 Hz and 0.25 millisecond respectively.

Q.2 Find Nyquist rate and Nyquist interval for solving analog signals and systems;

x(t) = 4sin(100πt) + 5Cos(4000πt) +7Sin(500πt)

Solution:- Firstly for x₁(t);

$w_1 = 1000 \pi \end.$

$\therefore 2 \pi f_1 = 1000 \pi \end.$

$\therefore f_1 = 500 Hz \end.$

Secondly for x₂(t);

$w_2 = 4000 \pi \end.$

$\therefore 2 \pi f_2 = 4000 \pi \end.$

$\therefore f_2 = 2000 Hz \end.$

Lastly for x₃(t);

$w_3 = 500 \pi \end.$

$\therefore 2 \pi f_3 = 500 \pi \end.$

$\therefore f_3 = 250 Hz \end.$

Therefore the highest frequency component = 2000 Hz.

$\therefore Nyquist ~ Rate = 2 * 2000 = 4000 Hz \end.$

$\therefore Nyquist ~ Interval = \frac {1} {4000} = 0.25 MilliSec \end.$

However, from the above solution, we can easily conclude that the value of Nyquist rate and Nyquist Interval is 4000 Hz and 0.25 MilliSecs respectively.

Even and Odd Signals

Even Signals

An Even Signal is a signal which is equal to its time reverse counter-part. Most certainly even signals are symmetrical about the vertical i.e. about the y axis. However, it is represented as;

$x(t) = x(-t)~ for ~ all ~ values ~ of ~ t\end.$

$x[n] = x[-n]~ for ~ all ~ values ~ of ~ n\end.$

Odd Signal

An Odd signal is a signal whose time reverse signal is not equal to the original signal. However, these signal passes through the origin i.e. they are symmetric about the origin. Besides, it is express as;

$x(t) = -x(-t)~ for ~ all ~ values ~ of ~ t\end.$

$x[n] = -x[-n]~ for ~ all ~ values ~ of ~ n\end.$

Derivation for Even and Odd signals

Firstly Assume; x_e(t) → Even part of x(t)
                x_o(t) → Odd part of x(t)

Thus after making the assumptions we can start with the numerical part

$x(t) = x_e(t) + x_o(t)__________________(1) \end.$

Certainly, by replacing t by -t we get;

$x(-t) = x_e(-t) + x_o(-t)__________________(2) \end.$

Adding and subtracting equation 1 from 2 we get;

$x(t) + x(-t) = 2x_e(t) \end.$

$\therefore x_e(t) = \frac {1} {2} [x(t) + x(-t) ]________(Even)\end.$

$x(t) - x(-t) = 2x_o(t) \end.$

$\therefore x_o(t) = \frac {1} {2} [x(t) - x(-t) ]________(Odd)\end.$

However, by using similar means, we can get equations for even and odd functions in the frequency domain which are;

$x_e[n] = \frac {1} {2} [x[n] + x[-n] ]________(Even)\end.$

$x_o[n] = \frac {1} {2} [x[n] - x[-n] ]________(Odd)\end.$

Properties of Addition and Multiplication in signals and systems

Firstly addition properties
- The Sum of 2 even functions is even and of two odd functions is odd.
- The Sum of even to odd function is neither even nor odd.
Secondly properties of multiplication
- The product of 2 even functions is even and of 2 odd functions is odd.
- product of even to odd or odd to even function is odd.

Note:- However if the function is entirely even its even part is same as of x(t) and its odd part is zero and vice versa.

Significance:-

Most certainly used in Fourier Series.

Deterministic and Random Signal

Deterministic Signals are those Signal which can be express Mathematically for example:- Sine, Cosine, Expo.

Random Signals are those signals which cannot be expressed mathematically for example- Noise.

Periodic and Non-Periodic signal

Periodic Signal

A Periodic Signal is a signal which repeats itself after a fixed interval of time. Their values can be obtained at any particular instant of time. Moreover, they consist of a particular pattern. condition for periodicity is;

x(t) = x(t+T)_________[for C.T]
x[n] = x[n+N]_________[for D.T]

For example Sine, Cosine, Sawtooth. Besides they can also be said as deterministic signals.

Note:- A D.T signal is said to be periodic only if it is the ratio of integers and if this condition is violated it is a Non-Periodic Signal. periodic and non periodic

Non-Periodic Signal

A Signal which does not repeat itself again and again after a fixed interval of time is called a Non-Periodic. It can also be named as a type of random signal. there’s no particular pattern present. Certainly in this type of signal, the value cannot be determined at any instant. it is express as;

x(t) ≠ x(t+T)_________[for C.T]
x[n] ≠ x[n+N]_________[for D.T]

Conclusion

However here we are at the end of the blog. The fundamentals of the basic signals will be continued in the next blog. I hope that you like it and have got all your answers. Besides if you have any doubt then please feel free to ask down below and I will also like to know the topic which you will read next on.

Have a nice day 🙂

Regards.

#continuous and discrete signals #Even and odd signal #periodic and non-periodic signal #random and deterministic signal #sampling theorem #types of signal #what is nyquist interval #what is nyquist rate #what is signals #what is systems 3types of system

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