y1_input_5pt

Figure 1: Comparison of input and output waveform from the 5pt averager filter. Click on either waveform to hear how it sounds. Assuming the A/D and D/A converters use a sampling rate of 8000 Hz. what frequency do you expect the tone to have?

While the frequency of the output tone remains unchanged, it has been modified by a complex scale factor. In order to determine this scale factor, the frequency response of the five point averager filter must be evaluated at the frequency of interest:

$$ H(\hat\omega)|_{\hat\omega=0.1\pi} = (\frac{sin(\hat\omega5/2)}{5sin(\hat\omega/2)})e^{-j2\hat\omega}|_{\hat\omega=0.1\pi} = 0.9040e^{-j\pi/5} $$ Therefore, $$ y_1=\mathfrak Re\{0.9040e^{-j\pi/5} 3e^{j0.1\pi m + \pi/4}\} = 2.712 cos(0.1\pi n + 0.05\pi) $$