Page 62 - Fundamentals of Communications Systems

2.14 Chapter Two

signal x(t) is distributed as a function of frequency. The units on a power spec-
trum are Watts/Hz. Because of this characteristic, two important characteristics
of the power spectral density are

S x ( f , T m ) ≥ 0 ∀ f (Power in a signal cannot be negative valued)
1 T m /2 2 ∞
P x (T m ) = |x(t)| dt = S x ( f , T m )df (2.57)
T m −T m /2 −∞

In practice, the actual values output by a spectrum analyzer test instrument
are closer to

1 f +B r /2
˜ 2
S x ( f , T m ) = |X T m (λ)| dλ W (2.58)
T m f −B r /2
where B r is denoted as the resolution bandwidth (see [Pet89] for a good dis-
cussion of spectrum analyzers). Signal analyzers are designed to provide the
same information for an engineer as signal and systems theory does within the
constraints of electronic measurement techniques.
Also when measuring electronic signals practicing engineers commonly use
the decibel terminology. Decibel notation can express the ratio of two quanti-
ties, i.e.,

P 1
P = 10 log dB (2.59)
P 2
In communication systems, we often use this notation to compare the power
of two signals. The decibel notation is used because it compresses the mea-
6
surement scale (10 = 60 dB). Also multiplies turn into adds, which is often
convenient in the discussion of electronic systems. The decibel notation can
also be used for absolute measurements if P 2 becomes a reference level. The
most commonly used absolute decibel scales are dBm (P 2 = 1 mw) and dBW
(P 2 = 1 W). In communication systems, the ratio between the signal power and
the noise power is an important quantity and this ratio is typically discussed
with decibel notation.

2.2.3 Bandwidth of Signals
Engineers often like to quantify complicated entities with simple parameteriza-
tions. This often helps in developing intuition and simplifies discussion among
different members of an engineering team. Since the frequency domain descrip-
tion of signals is not often easily parameterized, engineers often like to describe
signals in terms of a single number. Most often in communications engineering,
this single parameter is a description of the bandwidth. Bandwidth in commu-
nication engineering most often refers to the amount of positive frequency
spectrum that a signal occupies. Unfortunately, the bandwidth of a signal does