In a four channel data acquisition system, the input channels are selected sequentially for A-D conversion using a clock as shown in the figure. On each clock edge the multiplexer switches to next channel (folding back to i0 from i3) and the output is sampled and quantized. If the inputs are A = 3cos 2π8000t, B = 5cos 2π4000t, C = 6sin 2π 2000t, what is the minimum clock frequency for proper sampling of all the inputs ?

Explanation:

Minimum Clock Frequency for Proper Sampling:

The problem revolves around determining the minimum clock frequency required for proper sampling of the inputs in a four-channel data acquisition system. The inputs are provided as:

• A = 3cos(2π8000t)
• B = 5cos(2π4000t)
• C = 6sin(2π2000t)

To ensure proper sampling, the Nyquist criterion must be satisfied. The Nyquist criterion states that the sampling frequency should be at least twice the highest frequency present in the signal. This is essential to avoid aliasing, which can lead to distortion and incorrect data acquisition.

Step 1: Determine the Highest Frequency

For each input signal, the frequency can be extracted from the coefficient of t in the trigonometric function:

• A = 3cos(2π8000t): The frequency is 8000 Hz.
• B = 5cos(2π4000t): The frequency is 4000 Hz.
• C = 6sin(2π2000t): The frequency is 2000 Hz.

The highest frequency among the signals is 8000 Hz.

Step 2: Apply the Nyquist Criterion

According to the Nyquist criterion, the sampling frequency must be at least twice the highest frequency. Hence:

Minimum Sampling Frequency = 2 × Highest Frequency = 2 × 8000 Hz = 16000 Hz

Step 3: Consider the Four-Channel System

In a four-channel system, the clock frequency determines how quickly the multiplexer switches between the input channels. To properly sample all four channels, the sampling frequency for each channel must be maintained at the minimum required value.

Since the system sequentially switches between four channels, the clock frequency must be 4 × Minimum Sampling Frequency to ensure each channel is sampled at least at the Nyquist rate. Therefore:

Clock Frequency = 4 × 16000 Hz = 64000 Hz