Quantization of a Sine Wave

3-Bit Sine Sampling¶

The following example quantizes a sine wave with 3 bit. This bit-depth is too low to result in acceptable results, but it is well suited for demonstrating the effects that occur during quantization. The sine wave has an amplitude of $1$, a frequency of $f=200 \mathrm{Hz}$ at a sampling rate of $f=48 \mathrm{kHz}$:

No description has been provided for this image

Audio Results¶

Original sine wave with $f_0 = 200 \mathrm{Hz}$:

Quantized sine wave at 16 kHz:

At 3 bit, the quantized sine wave is heavily distorted. The distortions are apparantly harmonic.

Quantization Error¶

The quantization error $e$ is the difference between the input signal and the quantized signal:

$$ e = x-x* $$

PDF¶

The PDF of the quantization error is Laplace shaped:

Spectrum¶

The magintude spectrum clearly shows the harmonic nature of the distortion, with peaks at integer multiples of the input signal's frequency $f_0=200$.

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Cell In[8], line 8
      5 f = np.linspace(0,fs/2,int(N/2))
      7 plt.plot(f,abs(E),'gray')
----> 8 plt.plot(f,abs(X),'k')
     10 ax.set_yscale('log')
     11 ax.set_ylim([0.01,10e5])

File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/matplotlib/pyplot.py:3794, in plot(scalex, scaley, data, *args, **kwargs)
   3786 @_copy_docstring_and_deprecators(Axes.plot)
   3787 def plot(
   3788     *args: float | ArrayLike | str,
   (...)
   3792     **kwargs,
   3793 ) -> list[Line2D]:
-> 3794     return gca().plot(
   3795         *args,
   3796         scalex=scalex,
   3797         scaley=scaley,
   3798         **({"data": data} if data is not None else {}),
   3799         **kwargs,
   3800     )

File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/matplotlib/axes/_axes.py:1779, in Axes.plot(self, scalex, scaley, data, *args, **kwargs)
   1536 """
   1537 Plot y versus x as lines and/or markers.
   1538 
   (...)
   1776 (``'green'``) or hex strings (``'#008000'``).
   1777 """
   1778 kwargs = cbook.normalize_kwargs(kwargs, mlines.Line2D)
-> 1779 lines = [*self._get_lines(self, *args, data=data, **kwargs)]
   1780 for line in lines:
   1781     self.add_line(line)

File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/matplotlib/axes/_base.py:296, in _process_plot_var_args.__call__(self, axes, data, *args, **kwargs)
    294     this += args[0],
    295     args = args[1:]
--> 296 yield from self._plot_args(
    297     axes, this, kwargs, ambiguous_fmt_datakey=ambiguous_fmt_datakey)

File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/matplotlib/axes/_base.py:486, in _process_plot_var_args._plot_args(self, axes, tup, kwargs, return_kwargs, ambiguous_fmt_datakey)
    483     axes.yaxis.update_units(y)
    485 if x.shape[0] != y.shape[0]:
--> 486     raise ValueError(f"x and y must have same first dimension, but "
    487                      f"have shapes {x.shape} and {y.shape}")
    488 if x.ndim > 2 or y.ndim > 2:
    489     raise ValueError(f"x and y can be no greater than 2D, but have "
    490                      f"shapes {x.shape} and {y.shape}")

ValueError: x and y must have same first dimension, but have shapes (48000,) and (101,)