How to solve the Fast Fourier transform?

Amplitude modification, why do fourier transform of a signal always symmetrical about the vertical axis?

• Take the example on wikipedia. http://en.wikipedia.org/wiki/Amplitude_modulation this graph on that Wikipedia page => http://en.wikipedia.org/wiki/File:AM_spectrum.svg shows that fourier transform of signal M is symmetrical about the vertical axis. I have looked through other books, even in a general case where the signal has some crazy shape (i.e. definitely not an even function) , Fourier transform of that signal is always symmetrical about the vertical axis. (and hence when you perform amplitude modulation, you get double sideband.) I don't understand why does it have to be symmetrical. Is it a property of Fourier transform? If so, what is the reason behind it? Many thanks

• Answer:

  With AM as in medium wave broadcast radio, the sidebands are the same (except the frequency order is reversed), so in a spectrum display there is a carrier with an upper and lower sideband. As the sidebands are the same, the display is symmetrical. In practice they may be slightly different due to tuned circuits following the modulator not being exactly centered and symmetrical about the carrier. This comes from the modulation process: Upper sideband = carrier + modulation signal Lower sideband = carrier - modulation signal. Lets say the modulation signal and so bandwidth extends from 20 Hz upwards to 10kHz. The upper sideband corresponds to carrier + 20Hz to carrier + 10kHz. The lower sideband corresponds to carrier - 20Hz to carrier - 10kHz. The shape of the modulation band is reproduced in the envelope of each sideband, rather than the waveform. This is like a spectrum of the modulating signal. Thus a square wave displays as many odd harmonics tapering off at higher frequencies. Each bar in the histogram (bar graph) represents the power in a specific bandwidth, which is something like one half the frequency spacing to the next bar. It is a plot of the power density versus frequency, showing how a time series or signal is distributed with frequency. The fourier transform is the equivalent of many bandpass filters in parallel, one for each frequency bar that can be represented. If the sidebands are not symmetrical then by definition the signal is not AM double sideband. It is possible to have one single sideband, with or without the carrier, or even independent sidebands (different information on upper and lower sideband). With analogue AM television one sideband is partly suppressed by filters, so this is called vestigial sideband (meaning remains or remnant of a sideband).