(a) Explain the frequency mode, the period mode and the multiple period mode of measurement of a counterbaseddigital time/frequency/period measurement system. (8+7+5)(b) The time period of a sinusoidal signal having a frequency of 1kHz with a signal-to-noise ratio of 40dB ismeasured in (a) the frequency mode with a gate time of 1s, (b) the period mode with a clock resolution of100s, and (c) the multiple-period mode with 100 period measurement. The clock accuracy (f/f) is 10^-8.Justify with calculations as to which of the three measuring modes gives the highest accuracy. Assume, forsimplicity, the noise to be another sinusoid of frequency 10kHz.

(a) In a counter-based digital time/frequency/period measurement system, there are three modes of measurement:

1. Frequency Mode: In this mode, the number of cycles of the input signal that occur within a fixed gate time is counted. The frequency is then calculated by dividing the number of cycles by the gate time. This mode is most accurate when measuring high frequencies over a long gate time.

2. Period Mode: In this mode, the time it takes for one cycle of the input signal to occur is measured. This is done by counting the number of clock cycles that occur during one cycle of the input signal. The period is then calculated by multiplying the number of clock cycles by the clock period. This mode is most accurate when measuring low frequencies or when the gate time is short.

3. Multiple Period Mode: This mode is a variation of the period mode where the time for multiple cycles of the input signal is measured. The period is then calculated by dividing the total time by the number of cycles. This mode increases the measurement time, which can improve accuracy for low frequencies or noisy signals.

(b) To determine which mode gives the highest accuracy, we need to calculate the measurement error for each mode:

1. Frequency Mode: The measurement error is determined by the clock accuracy and the gate time. For a 1kHz signal with a gate time of 1s and a clock accuracy of 10^-8, the measurement error is 1kHz * 10^-8 = 10^-5Hz.

2. Period Mode: The measurement error is determined by the clock resolution and the signal period. For a 1kHz signal with a clock resolution of 100µs, the signal period is 1ms. The measurement error is 100µs / 1ms = 0.1.

3. Multiple Period Mode: The measurement error is determined by the clock accuracy, the signal period, and the number of periods measured. For a 1kHz signal with 100 period measurement and a clock accuracy of 10^-8, the signal period is 1ms. The measurement error is (1ms * 10^-8) / 100 = 10^-13.

Therefore, the multiple period mode gives the highest accuracy.