<< Chapter < Page | Chapter >> Page > |
Perhaps the best way to understand the Fourier Transform is to look closely at the inverse function
The complex exponential can be interpreted as a (complex valued) sinusoidal wave since it is the sum of asine term and a cosine term, both of frequency (via Euler's formula). Since is a complex number at each , [link] can be interpreted as describing or decomposing into sinusoidal elements of frequencies weighted by the . The discrete approximation to the Fourier transform,called the DFT, is discussed in some detail in [link] , and a table of useful properties appearsin [link] .
If, at any particular frequency , the magnitude spectrum is strictly positive( ), then the frequency is said to be present in . The set of all frequencies that are present in the signalis the frequency content , and if the frequency content consists only of frequencies below some given , then the signal is said tobe bandlimited to . Some bandlimited signals are
But real world signals are never completely bandlimited, and there is almost always some energy at every frequency.Several alternative definitions of bandwidth are in common use which try to capture theidea that “most of” the energy is contained in a specified frequency region.Usually, these are applied across positive frequencies, with the presumption that the underlying signalsare real valued (and hence have symmetric spectra). Here are some of the alternative definitions:
These definitions are illustrated in [link] .
The various definitions of bandwidth refer directly to the frequency content of a signal.Since the frequency response of a linear filter is the transform of its impulse response, bandwidth is alsoused to talk about the frequency range over which a linear system or filter operates.
TRUE or FALSE: Absolute bandwidth is never less than 3-db power bandwidth.
Suppose that a signal is complex-valued and hence has a spectrum that is not symmetric about zero frequency.State versions of the various definitions of bandwidth that make sense in this situation.Illustrate your definitions as in [link] .
Suppose that the signal contains important information that must be transmitted. There are many kinds ofoperations that can be applied to . Linear time invariant (LTI) operations are those for which superposition applies, but LTI operations cannot augment the frequencycontent of a signal—no sine wave can appear at the output of a linear operation if it was not already present inthe input.
Notification Switch
Would you like to follow the 'Software receiver design' conversation and receive update notifications?