Answer and Explanation:
AI-generated answer
i) If the information signals are sinusoidal and sampled at twice the Nyquist rate, the maximum information frequency is 10 kHz. The Nyquist rate states that the sampling rate should be at least twice the maximum frequency of the signal. Therefore, the sampling rate will be 20 kHz.
To find the number of messages that can be encoded, we need to calculate the available bandwidth per message. The channel bandwidth is 1 MHz, and we have 20 kHz for each message.
The number of messages that can be encoded is given by:
Number of messages = Channel bandwidth / Bandwidth per message
Number of messages = 1 MHz / 20 kHz
Number of messages = 50
Therefore, if the information signals are sinusoidal, we can encode up to 50 messages.
ii) If the information signals are non-sinusoidal with a crest factor of 6 and the probability of error is 10^-5, we can find the overall system signal-to-noise ratio (SNR).
The crest factor is the ratio of the peak amplitude of the signal to its root mean square (RMS) value. In this case, the crest factor is 6.
The probability of error is related to the SNR by the Q-function, which gives the probability that a Gaussian random variable exceeds a certain threshold. The relationship between the probability of error (Pe) and SNR is given by:
Pe = Q(sqrt(2 * SNR))
Given that the probability of error is 10^-5, we can solve for the SNR:
10^-5 = Q(sqrt(2 * SNR))
Taking the inverse Q-function, we can solve for sqrt(2 * SNR):
sqrt(2 * SNR) = Q^(-1)(10^-5)
Now, let's assume that the overall system SNR is SNR_total. The quantization noise is given as 1.3 mW.
SNR_total = (Peak power of the signal) / (RMS power of the quantization noise)
The RMS power of the quantization noise can be calculated as the square root of the quantization noise power:
RMS power of quantization noise = sqrt(1.3 mW)
Finally, we can calculate the overall system SNR:
SNR_total = (6^2) / (RMS power of quantization noise)^2
Please note that further calculations are needed using the given values to obtain the final numerical result for the overall system SNR.
In summary:
i) If the information signals are sinusoidal and sampled at twice the Nyquist rate, we can encode up to 50 messages.
ii) To find the overall system SNR for non-sinusoidal signals with a crest factor of 6 and a given probability of error, further calculations are needed using the provided values.