# Statistics of the output of a high bitrate signal filtered through a narrow band low pass filter

I was wondering what would be the statistics of the output of the filtering of an extremely wideband signal (e.g. 10 Gbps random data stream) through a narrowband low-pass filter (e.g. 10 MHz LP). I have the feeling that the statistics of the output would be somehow Gaussian due to the central limit theorem, but I cannot give a clear explanation to this.

• In filter with such a low cut-off frequency (in your example, around $\pi/1000$), an output sample is necessarily the mixing of many samples of the input signal. In the case of a FIR filter, you need lots of taps; for an IIR filter, you need poles very near the unit circle, and thus input samples survive for a long time inside the filter. Such heavy mixing (linear combination of many samples) suggests CLT validity. – Juancho May 17 '14 at 19:14
• That was somehow my explanation. Speaking of analog systems, the time constant of the filter will be much larger than a single time period: many bits will therefore take part in the convolution, making CLT hold. – Aglar May 17 '14 at 20:45