8 - 1x oversampling

Write a section on the Wiki about how a CD player works with no oversampling, but digital filtering (1x oversampling)

First we sample the data at $f_{s}={\frac {1}{T}}$ to get the data in digital form and when you use the impulse function in time the frequency repeats forever as shown

                                 Sample $f_{s}={\frac {1}{T}}\Longrightarrow$

Where the impulse function with respect to time = $\sum _{n=-\infty }^{\infty }x(nt)\delta (t-nT)$
and ${\frac {1}{T}}\sum _{n=-\infty }^{\infty }X(f-{\frac {n}{T}})$ in the frequency domain.

Now that we have the data in the computer we want to convolve it with another impulse function $\sum _{m=-M}^{M}h({\frac {mT}{n}})\delta (t-{\frac {mT}{n}})$ in the time domain. This is where more that 1x oversampling would occur replace n for nx oversampling, but in our case we use n = 1. Doing this convolution we can also shape our frequency when you convolve in time it multiplies in frequency. So we pick a frequency function $\sum _{m=-M}^{M}h({\frac {mT}{n}})e^{-j2\pi fm{\frac {T}{n}}}$ to multiply by so our frequency won't overlap later and help compensate for losses in the D/A converter. Example the frequency function is.

$\sum _{m=-M}^{M}h({\frac {mT}{n}})e^{-j2\pi fm{\frac {T}{n}}}\Longrightarrow$

Now doing the convolution/multiplication yields.
The time domain stays the same with

But frequency

x

=

>