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Sec. 6–4 Linear Systems 443
Example 6–8 SIGNAL-TO-NOISE RATIO AT THE OUTPUT
OF A RC LOW-PASS FILTER
Refer to Fig. 6–9 again, and assume that x(t) consists of a sine-wave (deterministic) signal plus
white ergodic noise. Thus,
and
where A 0 , v0, and u 0 are known constants and ni (f) = N 0 >2. The input signal power is
and the input noise power is
Hence, the input signal-to-noise ratio (SNR) is
x(t) = s i (t) + n i (t)
s i (t) = A 0 cos(v 0 t + u 0 )
q
q
8n 2 i 9 = n 2 i = ni (f) df =
L L
-q
8s i 2 (t)9 = A 0 2
2
-q
N 0
2
df =q
a S N b in
= 8s i 2 (t)9
8n i 2 (t)9 = 0
(6–92)
Because the system is linear, the output consists of the sum of the filtered input signal plus the
filtered input noise:
The output signal is
y(t) = s 0 (t) + n 0 (t)
and the output signal power is
s 0 (t) = s i (t) * h(t)
8s 2 0 (t)9 = A 0 2
2 ƒ H(f 2
0) ƒ
From Eq. (6–90) of Example 6–7, the output noise power is
= A 0 ƒ H(f 0 ) ƒ cos[v 0 t + u 0 + lH(f 0 )]
n 0 2 = N 0
4RC
(6–93)
(6–94)
The output SNR is then
a S N b out
= 8s 0 2 (t)9
8n 0 2 (t)9
= 8s 0 2 (t)9
n 2 0(t)
= 2A 0 2 ƒ H(f 0 ) ƒ 2 RC
N 0
(6–95)