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`PRENTICE HALL COMMUNICATIONS ENGINEERING AND EMERGING TECHNOLOGIES SERIES
`Theodore S. Rappaport, Series Editor
`
`(2560-1022
`
`Cisco v. TQ Delta
`Pa - e 1 of 5
`
`CSCO-1022
`Cisco v. TQ Delta
`Page 1 of 5
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`PRENTICE HALL COMMUNICATIONS ENGINEERING
`AND EMERGING TECHNOLOGIES SERIES
`
`Theodore S. Rappaport, Series Editor
`
`RAPPAPORT Wireless Communications: Principles and Practice
`
`RAZAVI RF Microelectronics
`
`Forthcoming:
`
`LIBERTI & RAPPAPORT CDMA and Adaptive Antennas for Wireless Systems
`TRANTER, KURT, KOSBAR, & RAPPAPORT Simulation of Modern Communications
`Systems with Wireless Applications
`GARG & WILKES Global System Mobile Communication
`
`
`
`Page 2 of 5
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`Page 2 of 5
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`
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`RF MICROELECTRONICS
`
`,
`
`Behzad Razavi
`
`University of California, Los Angeles
`
`http://www.prenhall.com/mail_lists
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`To join a Prentice Hall PTR
`Internet mailing list, point to
`
`PRENTICE HALL PTR
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`Upper Saddle River, NJ 07458
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`Page 3 of 5
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`Sec. 2.3 Random Processes and Noise
`
`33
`
`It is useful to remember that for a Gaussian distribution approximately 68% of
`the sampled values fall between m — 0 and m + 0 and 99% between m — 30 '
`and m + 30.
`
`Since our knowledge of random signals in the
`Power Spectral Density
`time domain is usually quite limited, it is often necessary to characterize such
`signals in the frequency domain as well.
`In fact, as we will see throughout
`this book, the frequency-domain behavior of random signals and noise proves
`much more useful in RF design than do their time-domain characteristics.
`For a deterministic signal x (t), the frequency information is embodied in
`the Fourier transform:
`
`+00
`
`X(f) = / x(t) exp(—j27rft)dt.
`
`—oo
`
`(2.63)
`
`While it may seem natural to use the same definition for random signals, we
`must note that the Fourier transform exists only for signals with finite energy,3
`+00
`
`(2.64)
`|x(t)|2dt < oo,
`E, = f
`i.e., only if |x(t)|2 drops rapidly enough as t —-> oo. As shown in Fig. 2.21, this
`condition is violated by two classes of signals: periodic waveforms and random
`signals. In most cases, however, these waveforms have a finite power:
`
`—oo
`
`1
`P = lim —
`T—>oo T _T/2
`
`+T/2
`
`|x(t)|2dt < oo.
`
`(2.65)
`
`For periodic signals with P < co, the Fourier transform can still be defined
`by representing each component of the Fourier series with an impulse in the
`frequency domain. For random signals, on the other hand, this is generally not
`possible because a frequency impulse indicates the existence of a deterministic
`sinusoidal component. Another practical problem is that even if we somehow
`define a Fourier transform for a random (stationary or nonstationary) process,
`the result itself is also a random process [5].
`
`2
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`Figure 2.21
`
`Signals with infinite energy.
`
`3 The definition of energy can be visualized if x (t) is a voltage applied across a 1-9 resistor.
`
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`Page 4 of 5
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`Page 4 of 5
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`
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`34
`
`Chap. 2 Basic Concepts in RF Design
`
`s.
`
`
`
`
`er that frequency-domain characteristics
`From the above discussion we inf
`of random signals are embodied in a function different from a direct Fourier
`transform. The power spectral density (PSD) (also called the “spectral density”
`orsimply the “spectrum") is such a function. Before giving a formal definition of
`PSD, we describe its meaning from an intuitive point of view [6]. The spectral
`density, S,. (f), of a random signal x(z) shows how much power the signal
`carries in a unit bandwidth around frequency f. As illustrated in Fig. 2.22,
`if we apply the signal to a bandpass filter with a l-Hz bandwidth centered at
`f and measure the average output power over a sufficiently longtime (on the
`order of 1 s), we obtain an estimate of S, (f). If this measurement is performed
`for each value of f, the overall spectrum of the signal is obtained. This is in
`fact the principle of operation of spectrum analyzers.‘
`
`Band-Pass
`Filters
`
`Power
`
`
`
`Figure 2.22 Measurement of spectrum.
`
`The formal definition of the PSD is as follows [3]:
`sx(f) = TlgI1m .
`
`(2.66)
`
`XT(f) =10 x(t)exp(—j2J'tft)dt.
`
`(2.67)
`
`
`
`
`
`____._{______
`
`idth and a center frequency of. say, 1 GHz is im-
`4 Building a low-loss BPF with l-I-I7. baudw
`anslate the spectrum to a lower center frequency
`practical. Thus, actual spectrum analyzers both tr
`and measure the power in a hand wider than 1 Hz.
`
` T
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`Page 5 of 5
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`Page 5 of 5
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