Please don't mind me elaborating a bit
Say,
bandwidth of the original signal is B, which is
sampled (@ fs) and then quantized for
A2D conversion:
-- if sampled at Nyquist rate:
fs = 2B
-- if over-sampled,
fs = N*2B (where N is the oversampling ratio)
Now, in either of the above 2 cases, the output
signal will be
bandlimited from -B to +B. But, the
quantization noise spectra is from
-fs/2 to + fs/2 : depending on the cases above, fs/2 may or may not be = B !
View attachment 107455
So why is the quantization noise always in that range?
First, you have indicated in-band noise in your diagram. I assume you don't mean thermal noise or some analog noise source to the input of the A/D. OF course, quantization noise is not thermal noise or analog noise at the input of the A/D. It is modeled as input noise - usually white spectrum (uncorrelated samples) with uniform PDF distribution which is true under certain conditions. I'm not totally sure if your question is "why doesn't the quantization noise land outside the range from -fs/2 to fs/2", or if your question is,"why is it spectrally flat within that range from -fs/2 to fs/2.
That white spectrum means it's flat, evenly distributed from -fs/2 to fs/2. To say that it should be distributed outside that range is to misunderstand frequency aliasing. The normalized digital frequency is θ=wTs. So the nyquist zone from -fs/2 < f < fs/2 corresponds to -pi < θ < pi. So a normalized digital frequency at pi/2 gives
cos(pi/2 n) = [ 1 0 -1 0 1 0 -1 .... ]
Notice that cos(-3pi/2 n) and cos(5pi/2 n) give the same result (because n is an integer - discrete). They are the same discrete time signals, indistinguishable aliases. Any digital frequency θ=pi/2+k*2pi, where k is an integer is an alias of pi/2. In un-normalized frequency domain, any frequency f+kfs is an alias of f.
So why would quantization noise fall in that range from -fs/2 to fs/2? The discrete time spectrum is aliased or repeated every k*fs Hz. The discrete time spectrum from -fs/2 to fs/2 IS the discrete time spectrum from fs/2 to 3fs/2 which IS the discrete time spectrum from 3fs/2 to 5fs/2..
The assumption that it's white noise inside that range is a little more complicated to pin down. But generally, it will not be spectrally flat if the input signal is constant, or periodic, or correlated to the sampling frequency, or.... Take for example, if the input signal is a constant DC. Then the quantized digital code will always be the same, the quantization error will also always be the same, and all of the quantization noise will be DC.