A recording trace r ( t ) is defined as r ( t ) = s ( t ) + n ( t ), where s ( t
ID: 1845707 • Letter: A
Question
A recording trace r(t) is defined as r(t) = s(t) + n(t), where s(t) =10sin(20?t) (i.e., a 10 Hz sine wave, given ?= 2?f) is the signal and n(t) = sin(120?t) (i.e., the 60 Hz electromagnetic noise) the noise.
(a) Is r(t) periodic? If so, what are the fundamental frequency in Hz and fundamental period in second? Sketch r(t) in one period.
(b) Calculate the average power of r(t). Is the energy of r(t) finite?
(c) Calculate the signal-to-noise ratio (SNR) of the recording r(t), write it in decibels
(dB).
(d) Write the harmonics of the 60 Hz electromagnetic noise in the general form ?k (t);
what are the 2nd- and 3rd-order harmonics?
what are the 2nd- and 3rd-order harmonics?
Explanation / Answer
a) T1 = period of s(t) = 1/10 = 0.1 s
T2 = period of n(t) =1/60 = 0.0166 s
T = period of resultat signal = L.C.M of T1 and T2 = 0.6 s
b) average power = (Vrms)^2
Vrms = root mean square Volatge of r(t) = sqrt (100^2 /2 + 1^2/2) = sqrt(50.5)
average power = (Vrms)^2 = 50.5
c) no = power of noise = (Vrms)^2 =
Vrms = sqrt (1^2/2) = sqrt(0.5)
power of noise = 0.5
So = signal power =Vrms^2
Vrms = sqrt(10^2/2) =sqrt( 50)
So =signal power = 50
signal-to-noise ratio (SNR) = So/no = 50/0.5 =100
SNR (db ) = 10log( signal-to-noise ratio) = 20 db