EE422
HW#18
Due Tuesday, April 6 (Happy Easter!)
1. Congratulations!! You and your roommate saved your pennies and just bought a new CD player from K-MartÒ . Its A/D converter has 8 bits. You like heavy metal but your roommate likes easy listening so you decide to calculate the SNR of both your favorite songs and play the one with the higher SNR.
a) Your favorite heavy metal song is Def Leppard's classic, "I Saw Mommy Kissing Satan's Claws", and can be represented by the signal, x(t)=2+7cos1000t volts. Calculate the average power of x(t) then calculate the SNR for this tune.
b) Your roommate's muzak favorite is Montavani's chart topper, "1001 Strings Play Music to Rinse and Spit By", which can be represented by the signal, x(t)=0.5 + 1/4cos1000t+1/3sin1000t volts. Calculate the signal power then calculate the SNR for this catchy riff.
c) Which song has the higher SNR?
d) A real life audio signal (i.e., your hearing) is limited to a maximum frequency of 16 KHz. According to Nyquist, what should the minimum sampling rate of your CD player be?
e) In actuality, your CD player (like all CD players) samples at fs = 44.1 KHz. Why do you think is this different from your answer to part d)?
2. In class today, we showed that an expression for the Mean-Square-Error due to quantizing is
where D is the dynamic range, S is the width of each quantizing region (we called this q in class), and N is the number of bits of our A/D system. This expression was derived based upon time averaging. We can also obtain this expression by using the Probability Theory that you learned in EE421!!! Consider the following graphical representation of the process of quantizing x(t) when it resides inside a quantizing band of width S:
Without loss of generality, we can assume that the bottom level in the above figure corresponds to 0 and that the top level has a value of S. This means that the value of the quantized signal (xq(t)) will be a constant value of S/2. Next, let us assume that the actual signal (x(t)) is a random variable which is uniformly distributed from 0 to S (i.e., can take on any value from 0 to S with equal probability).
a) Find and sketch the probability density function, fx (Hint: recall the uniform density function)
b) Find the Mean of x (Hint:
)
c) Now find the Mean of x2 (Hint: 
)
d) Recall that the quantized signal xq (t) =S/2 = a constant. Thus, the error due to quantizing is x-xq . Use this fact to obtain an expression for the MSE due to quantization by finding the mean of the square of the error = E[(x-xq )2 ]= E[x 2]-2E[xxq ]+E[xq2 ] (Hint: E[k]=k and E[kx]=kE[x] where k is a constant).
e) Finally, we have been assuming that the "best" place to assign the quantized signal is smack dab in the middle of the quantizing band (i.e., xq(t) = S/2). Repeat part d) and find the MSE if xq(t) = k where k is some constant between 0 and S.
f) Show that the minimum MSE is obtained when k=S/2