The relationship between probability function and PSD

[svensl]

Hello,

I have a question whether/what relationship there is between the probability function and the power spectral density.

For example: If we consider data converters. When quantizing a signal then we often assume that the introduced error is uniformly distributed between -q/2 and q/2. Here q is the quantization step. In sigma delta converters, this error is shaped by a noise transfer function, often 1-z^(-1). Thus subtracting two consecutive errors equals noise shaping.
When looking at it from two errors each having a uniform distribution with probability of 1, then after subtracting them, I get a triangular probability. My question is how I can relate this triangular probability to the noise shaping function 1-z^(-1). For example, what constitutes the attenuation.

Can anyone help me find a connection?

Thanks

 

[elmolla]

Re: probability and PSD

There is a little misconception here. The quantization noise is dealt with as a continuous variable, so you can't simply use the Z-transform which is used with discrete-time signals, only.

Just convert it to Laplace transform (substitute for each z with exp(jwT)) and you'll get the noise shaping function. Not the triangular one.

 

[bulx]

Re: probability and PSD

More basic thing is that probability and spectral component are two different aspects of a random signal. A random signal (the quantization noise in your case) can have a uniform probability distribution, but that says nothing about its spectral components. (You could have a very slow or very fast varying quantization noise, depending on the speed of conversion, but the probability distribution is unchanged). But power spectrum and correlation are connected by Fourier transform.

If you are filtering noise, its PDF will be unchanged if the noise is gaussian. Of course, its spectrum is changed by your filter.
-b