HW8

Problem Statement

Write a section describing how a CD player works with out oversampling but with digital filtering (1x oversampling)

Solution

CD players are used to play audio files and although these are limited we can look at any non periodic signal as periodic with an infinite period. We can represent this signal, x(t) as such in the time domain and X(f) in the frequency domain.

In order to take the signal and read it digitally we must sample it. This gives us data points creating a discrete function of time $x(nT)$, where $n$ is and integer and $T$ is the period between samples. We use a frequency of 44 kHz as it is twice the frequency at which a human can hear (i.e. 2 x 22 kHz) -- that means, $f_s=44$ kHz. Mathematically sampling is represented by multiplying x(t) with the delta function which results in a convolution with X(f).

Represented as: $\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x(nT)\delta (t-n{T}_0)$ and $\frac{1}{T}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}X(f-\frac{n}{T})$ respectively.