fundamentals of engineering supplied-reference handbook - Ventech!

FM (Frequency Modulation)
The phase deviation is
t
φ () t = k m( λ) dλ
radians.
F
∫
−∞
The frequency-deviation ratio is
kFmax m( t)
D =
2πW
where W is the message bandwidth. If D 1, (wideband FM) the 98% power bandwidth B is
given by Carson’s rule:
B ≅ 2( D+ 1) W
The complete bandwidth of an angle-modulated signal is
infinite.
A discriminator or a phase-lock loop can demodulate anglemodulated
signals.
Sampled Messages
A lowpass message m(t) can be exactly reconstructed from
uniformly spaced samples taken at a sampling frequency of
fs = 1/Ts
f ≥ 2 W where M( f ) = 0 for f > W
s
The frequency 2W is called the Nyquist frequency. Sampled
messages are typically transmitted by some form of pulse
modulation. The minimum bandwidth B required for
transmission of the modulated message is inversely
proportional to the pulse length τ.
1
B ∝
τ
Frequently, for approximate analysis
1
B ≅
2τ
is used as the minimum bandwidth of a pulse of length τ.
178
ELECTRICAL AND COMPUTER ENGINEERING (continued)
Ideal-Impulse Sampling
n=+∞ n=+∞
x () t = m() t δ( t− nT ) = m( nT ) δ( t −nT
)
δ
∑ ∑
s s s
n=−∞ n=−∞
k=+∞
X ( f) = M( f) ∗ f δ( f −kf
)
δ
k =+∞
∑
∑
s s
k=−∞
= f M( f −kf
)
s s
k =−∞
The message m(t) can be recovered from xδ (t) with an ideal
lowpass filter of bandwidth W.
PAM (Pulse-Amplitude Modulation)
Natural Sampling:
A PAM signal can be generated by multiplying a message
by a pulse train with pulses having duration τ and period
Ts = 1/fs
n=+∞ n=+∞
⎡t−nTs⎤ ⎡t −nTs⎤
xN() t = mt () ∑ Π ⎢ ⎥= ∑ mt () Π
τ
⎢
τ
⎥
⎣ ⎦ ⎣ ⎦
n=−∞ n=−∞
k =+∞
∑
X ( f) =τf sinc( kτf ) M( f −kf
)
N s s s
k =−∞
The message m(t) can be recovered from xN(t) with an ideal
lowpass filter of bandwidth W.
PCM (Pulse-Code Modulation)
PCM is formed by sampling a message m(t) and digitizing
the sample values with an A/D converter. For an n-bit
binary word length, transmission of a pulse-code-modulated
lowpass message m(t), with M(f) = 0 for f > W, requires the
transmission of at least 2nW binary pulses per second. A
binary word of length n bits can represent q quantization
levels:
q = 2 n
The minimum bandwidth required to transmit the PCM
message will be
B∝ nW =
2W log2q