To establish the amplitude reduction of a simple moving average on a sinusoid, you need to understand how the moving average works as a lowpass filter.
A simple moving average (SMA) is a method of smoothing data by calculating the average of a specific number of consecutive data points. When applied to a sinusoidal signal, the SMA acts as a lowpass filter by attenuating the higher frequency components of the signal.
The amplitude reduction caused by the SMA depends on the length of the averaging window and the frequency of the sinusoid. Let's break down the process step by step:
Sinusoidal signal: Assume you have a sinusoidal signal with a certain amplitude and frequency.
Averaging window: Choose a specific length for the averaging window. The length of the window determines how many data points are included in the calculation of the moving average. For example, if you have a window length of N, you will average the current data point with the previous N-1 data points.
Calculation of moving average: Calculate the moving average of the sinusoidal signal using the chosen window length. At each time step, sum the last N data points (including the current one) and divide the sum by N to obtain the moving average value.
Amplitude reduction: The amplitude reduction occurs because the moving average smoothes out the rapid fluctuations in the sinusoid, effectively reducing the amplitude of the high-frequency components. The extent of the reduction depends on the relationship between the frequency of the sinusoid and the window length of the moving average.
- If the frequency of the sinusoid is much lower than the reciprocal of the window length, the moving average will not significantly affect the sinusoid's amplitude.
- If the frequency of the sinusoid is close to or higher than the reciprocal of the window length, the moving average will attenuate the high-frequency components, leading to a reduction in amplitude.
In general, as the window length increases, the moving average has a stronger lowpass filtering effect, resulting in greater amplitude reduction for higher-frequency sinusoids.
It's worth noting that the moving average is a linear filter, and its frequency response exhibits a gradual roll-off rather than a sharp cutoff. Therefore, the amplitude reduction will not be instantaneous or completely eliminate the higher frequencies but gradually reduce their contribution to the output signal.
Keep in mind that the above explanation assumes an idealized moving average without considering any windowing functions or other modifications. In practice, different types of moving average filters may introduce additional effects on the amplitude reduction.