0.1 A telecommunication system  (Page 3/15)

Perhaps the best way to understand the Fourier Transform is to look closely at the inverse function

The complex exponential $e^{j2\pi ft}$ can be interpreted as a (complex valued) sinusoidal wave since it is the sum of asine term and a cosine term, both of frequency $f$ (via Euler's formula). Since $W\left(f\right)$ is a complex number at each $f$ , [link] can be interpreted as describing or decomposing $w\left(t\right)$ into sinusoidal elements of frequencies $f$ weighted by the $W\left(f\right)$ . 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] .

Bandwidth

If, at any particular frequency $f_0$ , the magnitude spectrum is strictly positive( $\left|W\right(f_0\left)\right|>0$ ), then the frequency $f_0$ is said to be present in $w\left(t\right)$ . 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 $f^{†}$ , then the signal is said tobe bandlimited to $f^{†}$ . Some bandlimited signals are

• Telephone quality speech – maximum frequency $\sim$ 4 KHz and
• Audible music—maximum frequency $\sim$ 20 KHz.

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:

1. Absolute bandwidth is the smallest interval $f_2-f_1$ for which the spectrum is zero outside of $f_1<f<f_2$ (only the positive frequencies need be considered).
2. 3-dB (or half-power) bandwidth is $f_2-f_1$ , where, for frequencies outside $f_1<f<f_2$ , $\left|H\right(f\left)\right|$ is never greater than $1/\sqrt{2}$ times its maximum value.
3. Null-to-null (or zero-crossing) bandwidth is $f_2-f_1$ , where $f_2$ is first null in $\left|H\right(f\left)\right|$ above $f_0$ and, for bandpass systems, $f_1$ is the first null in the envelope below $f_0$ where $f_0$ is frequency of maximum $\left|H\right(f\left)\right|$ . For baseband systems, $f_1$ is usually zero.
4. Power bandwidth is $f_2-f_1$ , where $f_1<f<f_2$ defines the frequency band in which 99% of the total power resides.Occupied bandwidth is such that 0.5% of power is above $f_2$ and 0.5% below $f_1$ .

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.

Upconversion at the transmitter

Suppose that the signal $w\left(t\right)$ contains important information that must be transmitted. There are many kinds ofoperations that can be applied to $w\left(t\right)$ . 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.