Fourier transform of a quantized signal

[mnb96]

Hello,
this time it's hard to tell whether this is the right forum to post this thread.

Suppose I have a continuous function $f:\mathbb{R}\rightarrow [0,100)$, whose Fourier transform exists and is known. Note that the codomain of the function is composed by all the real numbers between 0 and 100.

If I "quantize" the values f(x), how this quantization affects the spectral representation of f ?

More formally, suppose I have f(x) defined as above, and also its Fourier transform F(ω) is known.
I want to consider the function: $$g(x)=\lfloor f(x) \rfloor$$.
How can I study the Fourier transform G(ω) ?

 

[marcusl]

I am not sure whether you are quantizing the amplitude of f in the time domain, or evaluating f at discrete times. I'll assume you mean the latter: g(t) is the product of f(t) and a train or sum of delta functions at the discrete times of interest. The continuous spectrum G(w) is therefore the convolution of F(w) with the transform of your delta function train. If the train happens to consist of deltas at equally spaced times (a comb), then its transform is also a comb (a train of deltas at equally spaced frequencies). This latter case arises in signal processing as the first step in producing a discrete Fourier transform (DFT) as well as in explaining aliasing. It therefore has a vast literature that you can access by searching on "discrete time Fourier transform."

 

[mnb96]

Hi marcusl,

unfortunately I have to admit that I meant exactly the former of the two cases mentioned by you. I am interested in what happens when the "amplitude" in time domain of the function f(x) is quantized. I tried to express this when I wrote that I am interested in knowing whether it is possible to study (or at least say something) about the Fourier transform of another function g(x) defined in the following way:

$$g(x)=\lfloor f(x) \rfloor$$

If f(x) is a continuous function $f:\mathbb{R} \rightarrow [0,100)$ then g(x) will be a function $g:\mathbb{R} \rightarrow \{0,1,2,\ldots,99 \}$.
For example if we define $f(x)=x$, then g(x) will look like a "staircase".

 

[marcusl]

Sorry, I'm not a mathematician, don't know what brackets missing their tops mean, and got confused over which axis the range refers to. You are in luck, however, because this problem has also been studied exhaustively since our digital world is powered by analog-to-digital converters (ADC's). Colored noise due to the quantization errors will add onto G(ω). The noise is non-white because the distribution of errors is non-Gaussian, assuming that the input signal takes on all real values. Pay attention to whether your quantizer rounds, or takes floor or ceiling values, in case it matters. (I can't remember if it does.)

Take a look at digital signal processing books like Oppenheim and Schafer, Discrete Time Signal Processing. If they don't discuss the spectrum adequately there, you will find what you need in the vast engineering literature. Folks at the Electrical Engineering forum here can also help you further with questions.

EDITere is an online reference that will get you started.
http://oldweb.mit.bme.hu/books/quantization/spectrum.pdf