To determine the number of bits per sample for a PCM signal, we need to consider the compression parameter and the minimum signal to quantization-noise ratio.

Compression Parameter:
The compression parameter determines the dynamic range of the PCM signal. It is defined as the ratio of the maximum amplitude of the signal to the minimum amplitude that can be represented. In this case, the compression parameter is given as 100.

Minimum Signal to Quantization-Noise Ratio:
The minimum signal to quantization-noise ratio (SQNR) is a measure of the signal quality. It is defined as the ratio of the signal power to the quantization noise power. The SQNR is typically specified in decibels (dB). In this case, the minimum SQNR is given as 50 dB.

Calculating the Number of Bits per Sample:
To calculate the number of bits per sample, we need to determine the maximum value that can be represented and the quantization step size.

Maximum Value:
The maximum value that can be represented for a PCM signal is determined by the compression parameter. In this case, the compression parameter is 100, so the maximum value is 100 times the minimum amplitude that can be represented.

Quantization Step Size:
The quantization step size is determined by the minimum SQNR. The SQNR in dB can be converted to a linear scale by taking the antilogarithm.

SQNR(dB) = 10 * log10(SQNR(linear))

50 dB = 10 * log10(SQNR(linear))
5 = log10(SQNR(linear))
SQNR(linear) = 10^5

The quantization step size (Δ) is given by:

Δ = maximum value / (2^N)

where N is the number of bits per sample.

Substituting the values:

Δ = (100 * minimum amplitude) / (2^N)

From the equation, we can see that the quantization step size is inversely proportional to the number of bits per sample. So, for a given compression parameter and minimum SQNR, we need to find the minimum number of bits per sample that satisfies the equation.

Finding the Number of Bits per Sample:
To find the number of bits per sample, we can start with a trial value and check if it satisfies the equation. If it does not, we increase the number of bits and repeat the process until we find the minimum number of bits that satisfies the equation.

Let's start with option 'A', which has 8 bits per sample:

Δ = (100 * minimum amplitude) / (2^8)
Δ = (100 * minimum amplitude) / 256

Now, we compare the quantization step size with the minimum value that can be represented (Δ <= minimum="" amplitude="">

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Let's calculate the quantization step size for option 'A':

Δ = (100 * minimum amplitude) / 256

If this value is less than or equal to the minimum amplitude divided by 2, then option 'A' is valid. Otherwise, we need to try a higher number of bits.

Performing the calculation and comparison, we find that option 'A'