When a signal contains noise, the average value and the amplitude of the zero frequency term can indeed be different. This discrepancy arises due to the nature of noise and its impact on the signal.
Noise is typically considered as random variations or unwanted disturbances that get added to the original signal. It can introduce fluctuations and irregularities in the signal's amplitude over time. When calculating the average value of the signal, these fluctuations, which are typically centered around zero, tend to average out to some extent, resulting in a value close to zero for the average.
On the other hand, the zero frequency term refers to the DC component or the constant offset present in the signal. It represents the average value of the signal without considering the noise. While the noise contributes to the fluctuations around this average value, the DC component or zero frequency term remains unaffected by the noise.
To illustrate this concept, consider a simple example. Let's say we have a signal that represents a pure sine wave with an amplitude of 1 unit and a frequency of 1 Hz. However, this signal is contaminated with noise, which introduces random fluctuations around the sine wave.
The average value of the signal, calculated over a sufficiently long period, would tend to be close to zero due to the random nature of the noise. However, the amplitude of the zero frequency term, which represents the pure sine wave without noise, would remain at its original value of 1 unit.
In summary, noise introduces random variations that affect the average value of the signal, causing it to approach zero. However, the zero frequency term, representing the pure signal without noise, retains its original amplitude as it remains unaffected by the noise.