# User:Caswto

9-27-2004

Todd Caswell
509-374-2820

### How a CD Player Works

First, a digital signal ${\displaystyle \ x(kt)}$ is read from the CD and then convolved with a pulse function ${\displaystyle \ p(t)}$ in the D/A converter. The result in the time domain looks like this:

${\displaystyle {\hat {x}}(t)=\sum _{k=-\infty }^{\infty }x(kT)p(t-kT)=p(t)*\sum _{k=-\infty }^{\infty }x(kT)\delta (t-kT)}$

Let's look at this result in frequency space. Note that convolution in time means multiplication in frequency.

${\displaystyle {\hat {X}}(f)=1/T\sum _{n=-\infty }^{\infty }X(f-n/T)\cdot P(f)}$

where

${\displaystyle P(f)=\int _{-T/2}^{T/2}e^{j2\pi ft}\,dt=Tsinc(fT)}$

The low pass filter then knocks the high frequencies out of the signal coming from the D/A converter, which smoothes out the edges of the reproduced sine wave ${\displaystyle {\hat {x}}(t)}$ in time. This output waveform then drives the speaker, thereby recreating the original sound stored on the CD.

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