Introduction:
The envelope detector is a simple and commonly used demodulation technique for extracting the message signal from an amplitude modulated (AM) signal. In order to successfully demodulate the signal using this technique, certain conditions must be met. One of these conditions is that the minimum value of the carrier signal amplitude (Ac) should be at least 2.

Explanation:
The amplitude modulated (AM) signal can be expressed as Ac cos(ωc t) * [1 + m cos(ωm t)], where Ac is the carrier signal amplitude, ωc is the carrier frequency, m is the modulation index, and ωm is the message signal frequency.

The envelope detector works by rectifying the AM signal to remove the negative half cycles, and then low-pass filtering the rectified signal to obtain the envelope of the original message signal.

Rectification:
During rectification, the negative half cycles of the AM signal are converted to positive half cycles. This is done by passing the AM signal through a diode, which allows current flow only in one direction.

Low-pass filtering:
After rectification, the signal contains both the envelope and high-frequency components. To extract the envelope, a low-pass filter is used to remove the high-frequency components. The cutoff frequency of the low-pass filter should be chosen such that it allows the message signal to pass through while attenuating the carrier frequency and its harmonics.

The minimum value of Ac should be at least 2 in order to ensure proper demodulation using the envelope detector.
Reason:
The reason behind this is that during rectification, the negative half cycles of the AM signal are clipped to zero. If the minimum value of Ac is less than 2, there is a possibility that the negative half cycles of the AM signal will be completely clipped, resulting in a distorted envelope. This distortion can lead to inaccurate demodulation and loss of the original message signal.

By setting the minimum value of Ac to 2 or higher, we ensure that the negative half cycles of the AM signal are not completely clipped, and the envelope can be accurately extracted using the envelope detector.

Conclusion:
In conclusion, the minimum value of the carrier signal amplitude (Ac) should be at least 2 in order to successfully demodulate the signal using the envelope detector. This is necessary to prevent distortion of the envelope and ensure accurate extraction of the message signal.

We have x_AM = A_c cos ω_ct + 2 cosω_m t cosω_ct