Found 189 Articles for Signals and Systems

The Fourier series can be used to analyse only the periodic signals, while the Fourier transform can be used to analyse both periodic as well as non-periodic functions. Therefore, the Fourier transform can be used as a universal mathematical tool in the analysis of both periodic and aperiodic signals over the entire interval. The Fourier transform of periodic signals can be found using the concept of impulse function.Now, consider a periodic signal $\mathit{x\left(t\right )}$ with period $\mathit{T}$. Then, the expression of $\mathit{x\left(t\right )}$ in terms of exponential Fourier series is given by, $$\mathrm{\mathit{x\left(t\right)=\sum_{n=-\infty }^{\infty } C_{n}\:e^{jn\omega _{\mathrm{0}}t}}}$$Where $\mathit{C_{n}}$ be the ... Read More

Linear Time-Invariant SystemA system for which the principle of superposition and the principle of homogeneity are valid and the input/output characteristics do not change with time is called the linear time-invariant (LTI) system.Impulse Response of LTI SystemWhen the impulse signal is applied to a linear system, then the response of the system is called the impulse response. The impulse response of the system is very important for understanding the behaviour of the system.Therefore, if$$\mathrm{\mathit{\mathrm{Input}, x\left(t\right)=\delta\left(t\right)}}$$Then, $$\mathrm{\mathit{\mathrm{Output}, y\left(t\right)=h\left(t\right)}}$$As the Laplace transform and Fourier transform of the impulse function is given by, $$\mathrm{\mathit{L\left [\delta\left(t\right) \right ]\mathrm{=}\mathrm{1}\:\:\mathrm{and} \:\:F\left [\delta\left(t\right) \right ]\mathrm{=}\mathrm{1}}}$$Hence, once the ... Read More

The transfer function of a continuous-time LTI system may be defined using Laplace transform or Fourier transform. Also, the transfer function of the LTI system can only be defined under zero initial conditions. The block diagram of a continuous-time LTI system is shown in the following figure.Transfer Function of LTI System in Frequency DomainThe transfer function 𝐻(𝜔) of an LTI system can be defined in one of the following ways −The transfer function of an LTI system is defined as the ratio of the Fourier transform of the output signal to the Fourier transform of the input signal provided that ... Read More

Condition of CausalityA causal system is the one which does not produce an output before the input is applied. Therefore, for an LTI (Linear Time-Invariant) system to be causal, the impulse response of the system must be zero for t less than zero, i.e., $$\mathrm{\mathit{h\left ( t \right )\mathrm{=}\mathrm{0};\; \; \mathrm{for}\: \: t< 0}}$$The term physical realization denotes that it is physically possible to construct that system in real time. A system which is physically realizable cannot produce an output before the input is applied. This is called the condition of causality for the system.Therefore, the time domain criterion for ... Read More

Fourier TransformFor a continuous-time function $\mathrm{\mathit{x\left ( t \right )}}$ , the Fourier transform of $\mathrm{\mathit{x\left ( t \right )}}$ can be defined as, $$\mathrm{\mathit{X\left ( \omega \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt }}$$And the inverse Fourier transform is defined as, $$\mathrm{\mathit{x\left ( t \right )\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X\left ( \omega \right )e^{j\omega t}d\omega }} $$Parseval’s Theorem of Fourier TransformStatement – Parseval’s theorem states that the energy of signal $\mathrm{\mathit{x\left ( t \right )}}$ [if $\mathrm{\mathit{x\left ( t \right )}}$ is aperiodic] or power of signal $\mathrm{\mathit{x\left ( t \right )}}$ [if $\mathrm{\mathit{x\left ( t \right )}}$ ... Read More

Distortion-less TransmissionWhen a signal is transmitted through a system and there is a change in the shape of the signal, it called the distortion. If the output of the system is an exact replica of the input signal, then the transmission of the signal through the system is called distortionless transmission.Linear Phase SystemFor distortionless transmission through a system, there should not be any phase distortion, i.e., the phase of the system should be linear. For the linear phase system, the impulse response of the system is symmetrical about the delay time $\mathit{(t_{d})}$.ProofFor a linear phase system, we have, $$\mathrm{ \mathit{H\left ... Read More

For a continuous-time function $\mathit{x(t)}$, the Fourier transform of $\mathit{x(t)}$ can be defined as$$\mathrm{\mathit{X\left ( \omega \right )\mathrm{\mathrm{=}}\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt}}$$And the inverse Fourier transform is defined as, $$\mathrm{\mathit{F^{\mathrm{-1}}\left [ X\left ( \omega \right ) \right ]\mathrm{\mathrm{=}}x\left ( t \right )\mathrm{\mathrm{=}}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X\left ( \omega \right )e^{j\omega t}d\omega }}$$Multiplication Property of Fourier TransformStatement – The multiplication property of continuous-time Fourier transform (CTFT) states that the multiplication of two functions in time domain is equivalent to the convolution of their spectra in the frequency domain. The multiplication property is also called frequency convolution theorem of Fourier ... Read More

For a continuous-time function $\mathit{x(t)}$, the Fourier transform of $\mathit{x(t)}$ can be defined as, $$\mathrm{\mathit{X\left(\omega\right )\mathrm{=}\int_{-\infty }^{\infty} x\left(t\right)\:e^{-j\omega t}\:dt} }$$Fourier Transform of Gaussian SignalGaussian Function - The Gaussian function is defined as, $$\mathrm{\mathit{g_{a}\left(t\right)\mathrm{=} e^{-at^{\mathrm{2}}} ;\:\:\mathrm{for\:all} \:t} }$$Therefore, from the definition of Fourier transform, we have, $$\mathrm{\mathit{X\left(\omega\right)\mathrm{=} F\left [e^{-at^\mathrm{2}} \right ]=\int_{-\infty }^{\infty}e^{-at^\mathrm{2}} \:e^{-j\omega t} \:dt}}$$$$\mathrm{\Rightarrow \mathit{X\left(\omega\right) \mathrm{=}\int_{-\infty}^{\infty} e^{-\left(at^\mathrm{2}+j\omega t\right) }\:dt \mathrm{=}e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)}\int_{-\infty}^{\infty}e^{\left [{-t\sqrt{a}+(j\omega/\mathrm{2}\sqrt{a})}\right]^{2}}}dt }$$Let, $$\mathrm{\mathit{\left [t\sqrt{a}+(j\omega /\mathrm{2}\sqrt{a})\right ]\mathrm{=} u}}$$Then, $$\mathrm{\mathit{du\mathrm{=} \sqrt{a} \:dt\:\mathrm{and}\: \:dt\mathrm{=} \frac{du}{\sqrt{a}}}}$$$$\mathrm{\mathit{\therefore X\left(\omega\right)\mathrm{=}e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)}\int_{-\infty }^{\infty} \frac{e^{-u^{\mathrm{2}}}}{\sqrt{a}}\:du\:\mathrm{=} \frac{e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)}}{\sqrt{a}}\int_{-\infty }^{\infty}e^{-u^{\mathrm{2}}} \:du}}$$$$\mathrm{\mathit{\because\int_{-\infty }^{\infty}e^{-u^{\mathrm{2}}} \:du\mathrm{=} \sqrt{\pi}}}$$$$\mathrm{\mathit{\therefore X\left(\omega\right)\mathrm{=}\frac{e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)}}{\sqrt{a}}\cdot \sqrt{\pi}\mathrm{=} \sqrt{\frac{\pi}{a}} \cdot e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)} } }$$Therefore, the Fourier transform of the Gaussian function is, $$\mathrm{\mathit{F\left [e^{-at^{\mathrm{2}}}\right ] \mathrm{=}\sqrt{\frac{\pi}{a}} ... Read More

Power Spectral DensityThe distribution of average power of a signal in the frequency domain is called the power spectral density (PSD) or power density (PD) or power density spectrum. The power spectral density is denoted by $\mathit{S\left (\omega \right )}$ and is given by, $$\mathrm{\mathit{S\left (\omega \right )\mathrm{=}\lim_{\tau \rightarrow \infty }\frac{\left | X\left (\omega \right ) \right |^{\mathrm{2}}}{\tau }}}$$AutocorrelationThe autocorrelation function gives the measure of similarity between a signal and its time-delayed version. The autocorrelation function of power (or periodic) signal $\mathit{x\left ( t \right ) }$ with any time period T is given by, $$\mathrm{\mathit{R\left(\tau \right)=\lim_{T\rightarrow \infty }\mathrm{\frac{1}{\mathit{T}}}\int_{-\left(T/\mathrm{2}\right)}^{T/\mathrm{2}}x\left(t\right)\:x^{*}\left(t-\tau \right)\:dt}}$$Where, ... Read More

What is a Filter?A filter is a frequency selective network, i.e., it allows the transmission of signals of certain frequencies with no attenuation or with very little attenuation and it rejects all other frequency components.What is an Ideal Filter?An ideal filter is a frequency selective network that has very sharp cut-off characteristics, i.e., it transmits the signals of certain specified band of frequencies exactly and totally rejects the signals of frequencies outside this band. Therefore, the phase spectrum of an ideal filter is linear.Ideal Filter CharacteristicsBased on the frequency response characteristics, the ideal filters can be of following types −Ideal ... Read More