The signaling rate of a time division multiplexing (TDM) signal for M input channels is given by:

Explanation:

Signaling Rate of Time Division Multiplexing (TDM) Signal

Definition: Time Division Multiplexing (TDM) is a digital multiplexing technique used to combine multiple data streams or signals into one stream by assigning each signal a different time slot in the sequence. The signaling rate of a TDM signal refers to the rate at which the multiplexed data is transmitted over the communication channel. This rate depends on the number of input channels (M), the sampling frequency (fs), and the bandwidth of the input signals (W).

Explanation of the Correct Option:

The correct option is:

Option 1: r = Mfs ≥ 2MW

In a TDM system, each input channel has a specific sampling rate (fs). According to the Nyquist sampling theorem, the minimum sampling rate for any signal must be at least twice its highest frequency component (2W) to ensure accurate signal reconstruction. Thus, for M input channels, the total signaling rate (r) must satisfy the condition:

r = M × fs, where fs ≥ 2W

Therefore:

r = M × fs ≥ 2M × W

This equation ensures that the TDM system operates effectively without aliasing and that all the input signals are sampled and transmitted correctly. The signaling rate must be at least 2M × W to accommodate all M channels, as each channel requires a bandwidth of W and a sampling rate of at least 2W.

Key Points:

• M: Number of input channels in the TDM system.
• fs: Sampling frequency of each channel, which must be at least twice the bandwidth (2W) to satisfy the Nyquist criterion.
• W: Bandwidth of each input channel.
• The total signaling rate (r) ensures that all channels are multiplexed and transmitted without loss of information.

Conclusion: The signaling rate of a TDM signal for M input channels is given by r = Mfs ≥ 2MW, as stated in Option 1. This ensures that the system adheres to the Nyquist sampling theorem and can effectively transmit all input signals.

Additional Information:

To further understand the analysis, let’s evaluate the other options:

Option 2: r = Mfs ≤ 2MW

This option is incorrect because the signaling rate cannot be less than or equal to 2MW. The condition r = Mfs ≥ 2MW ensures that the Nyquist criterion is satisfied and that all input signals are accurately sampled and transmitted. If r ≤ 2MW, the system would not meet the minimum requirements for sampling and could result in aliasing or loss of information.

Option 3: r = Mfs ≤ MW

This option is incorrect as it violates the Nyquist sampling theorem. If the signaling rate is less than or equal to MW, the sampling frequency (fs) for each channel would be less than 2W, leading to aliasing and inaccurate signal reconstruction. Such a scenario is not feasible for a TDM system.

Option 4: r = Mfs ≥ MW

While this condition might seem plausible at first glance, it is insufficient to ensure accurate signal reconstruction for all channels. The Nyquist theorem requires the sampling frequency to be at least twice the bandwidth (fs ≥ 2W). Therefore, the signaling rate must satisfy r = Mfs ≥ 2MW, not just MW, to avoid aliasing and ensure proper operation.

Conclusion:

Option 1 (r = Mfs ≥ 2MW) is the correct choice as it ensures that the TDM system adheres to the Nyquist criterion, allowing accurate sampling and transmission of all input signals. The other options fail to meet this critical requirement and would result in improper operation of the TDM system.