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Signal Through System in Time and Frequency

SignalSystemTF.gif

The Fourier transform has a wide variety of applications. In particular, it is a powerful tool for the analysis in signals going through systems, as illustrated in the figure above. We see that an LTI system can be described in time domain by its impulse response function $h(t)$, so that the system's response to a given input $x(t)$ can be obtained as a convolution

\begin{displaymath}y(t)=h(t)*x(t) \end{displaymath}

or in frequency domain by its frequency response function $H(\omega)={\cal F}[h(t)]$, so that its response is a simpler product

\begin{displaymath}Y(\omega)=H(\omega)X(\omega) \end{displaymath}

where $Y(\omega)={\cal F}[y(t)]$, $H(\omega)={\cal F}[h(t)]$ and $X(\omega)={\cal F}[x(t)]$ are the Fourier transforms of $y(t)$, $h(t)$ and $x(t)$, respectively.

Although both the forward and inverse Fourier transforms are needed for the frequency domain method, we will gain some benefits not possible in time domain. Most obviously, the response of an LTI system to an input $x(t)$ can be much more conveniently obtained in frequency domain by a multiplication, instead of the corresponding convolution in time domain. Moreover, in some applications, it may only be possible to carry out certain system analysis and design task in frequency domain, such as designing a system so that it will produce a desired response $y(t)$ to a certain input $x(t)$. In time domain, given $x(t)$ and $y(t)$, it is difficult to obtain the impulse response function $h(t)$ that satisfies $y(t)=h(t)*x(t)$. However, in frequency domain, given $X(\omega)$ and $Y(\omega)$, it is relatively straight forward to find the required frequency response function $H(\omega)$ by a simple division $H(\omega)=Y(\omega)/X(\omega)$.


next up previous
Next: About this document ... Up: Fourier Transforms and Signal Previous: Orthogonal transform as rotation
Ruye Wang 2010-01-16