Fourier Analysis and Signal Filtering
Non-sinusoidal periodic signals are made up of many discrete sinusoidal frequency components (see applet Fourier Synthesis of Periodic Waveforms). The process of obtaining the spectrum of frequencies H(f) comprising a time-dependent signal h(t) is called Fourier Analysis and it is realized by the so-called Fourier Transform (FT). Typical examples of frequency spectra of some simple periodic signals composed of finite or infinite number of discrete sinusoidal components are shown in the figure below.
However, most electronic signals are not periodic and also have a finite duration. A single square pulse or an exponentially decaying sinusoidal signal are typical examples of non-periodic signals, of finite duration. Even these signals are composed of sinusoidal components but not discrete in nature, i.e. the corresponding H(f) is a continuous function of frequency rather than a series of discrete sinusoidal components, as shown in the figure below.
H(f) can be derived from h(t) by employing the Fourier Integral:
This conversion is known as (forward) Fourier Transform (FT). The inverse Fourier Transform (FT-1) can also be carried out. The relevant expression is:
These conversions (for discretely sampled data) are normally done on a digital computer and involve a great number of complex multiplications (N2, for N data points). Special fast algorithms have been developed for accelerating the overall calculation, the most famous of them being the Cooley-Tukey algorithm, known as Fast Fourier Transform (FFT). With FFT the number of complex multiplications is reduced to Nlog2N. The difference between Nlog2N and N2 is immense, i.e. with N=106 , it is the difference between 0.1 s and 1.4 hours of CPU time for a 300 MHz processor.
All FT algorithms manipulate and convert data in both directions, i.e. H(f) can be calculated from h(t) and vice versa, or schematically:
Signal Smoothing Using Fourier Transforms
Selected parts of the frequency spectrum H(f) can easily be subjected to piecewise mathematical manipulations (attenuated or completely removed). These manipulations result into a modified or "filtered" spectrum HÌ(f). By applying FT-1 to HÌ(f) the modified signal or "filtered" signal hÌ(t) can be obtained. Therefore, signal smoothing can be easily performed with removing completely the frequency components from a certain frequency and up, while the useful (information bearing) low frequency components are retained. The process is depicted schematically below (the pink area represents the range of removed frequencies):
With this applet you can observe the frequency spectrum H(f) of certain signals h(t) and perform signal filtering operations. These signals consist of 512 data points and they are grouped as "Fixed", "Periodic" and "Impulse" type. Each of these signals can be selected by clicking on the corresponding radiobutton. h(t) and H(f) appear on two separate chart areas. The frequency of the periodic signals can be altered by the scrollbar labeled "Frequency". Similarly, the width of impulse signals can be adjusted by the scrollbar labeled "Width".
The user can also draw a signal of his/her choice by clicking the "User pen" radiobutton and moving up/down the vertical scrollbar acting like a pen. Soon after the completion of the recording on the h(t) chart, the frequency spectrum of the recorded signal appears.
Random normal noise of variable amplitude can be added in a similar way on any of the selected signals by operating on the scrollbar labeled "Noise".
The user can remove an adjustable range of low and/or high frequency components from the frequency spectrum H(f) to obtain a modified spectrum HM(f). This action can be performed by operating the "Lower frequency limit" and "Upper frequency limit" scrollbars. During this operation the corresponding modified signal hM(t) is displayed almost instantaneously, whereas the original signal h(t) is also projected in faint red color for comparison.
It is of interest to note the widening of the frequency spectrum upon decreasing the width of the impulse type signals. This indicates that sharp impulse signals are rich in high frequency components. Therefore a wide band amplifier is required for handling this type of signals.
It is also of interest to perform smoothing of noisy signals by cutting-off the high frequency components. Observe how the original signal is distorted when the cut-off frequency is shifted to lower frequency values.
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