y1_output_1st

Figure 1: Comparison of input and output waveform from the 1st difference 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 first difference filter must be evaluated at the frequency of interest:

$$ H(\hat\omega)|_{\hat\omega = 0.1\pi} = 1-e^{-j\hat\omega}|_{\hat\omega=0.1\pi} = 0.3129e^{j0.45\pi} $$ Therefore, $$ y_1 = \mathfrak Re\{0.3129e^{j0.45\pi} 3e^{j0.1\pi n + \pi/4}\} = 0.9387 cos(0.1\pi n + 0.7\pi) $$