The quantized signal as the output of a quantizer can be expressed with the quantization error \(e[k]\) as

\[x_Q[k] = \mathcal{Q} \{ x[k] \} = x[k] + e[k]\]

According to the introduced model, the quantization noise can be modeled as uniformly distributed white noise. Hence, the noise is distributed over the entire frequency range. The basic concept of noise shaping is a feedback of the quantization error to the input of the quantizer. This way the spectral characteristics of the quantization noise can be changed, i.e. spectrally shaped. Introducing a generic filter \(h[k]\) into the feedback loop yields the following structure

The quantized signal can be deduced from the block diagram above as

\[x_Q[k] = \mathcal{Q} \{ x[k] - e[k] * h[k] \} = x[k] + e[k] - e[k] * h[k]\]

where the additive noise model from above has been introduced and it has been assumed that the impulse response \(h[k]\) is normalized such that the magnitude of \(e[k] * h[k]\) is below the quantization step \(Q\). The overall quantization error is then

\[e_H[k] = x_Q[k] - x[k] = e[k] * (\delta[k] - h[k])\]

The power spectral density (PSD) of the quantization with noise shaping is calculated to

\[\Phi_{e_H e_H}(\mathrm{e}^{\,\mathrm{j}\,\Omega}) = \Phi_{ee}(\mathrm{e}^{\,\mathrm{j}\,\Omega}) \cdot \left| 1 - H(\mathrm{e}^{\,\mathrm{j}\,\Omega}) \right|^2\]

Hence the PSD \(\Phi_{ee}(\mathrm{e}^{\,\mathrm{j}\,\Omega})\) of the quantizer without noise shaping is weighted by \(| 1 - H(\mathrm{e}^{\,\mathrm{j}\,\Omega}) |^2\). Noise shaping allows a spectral modification of the quantization error. The desired shaping depends on the application scenario. For some applications, high-frequency noise is less disturbing as low-frequency noise.

If the feedback of the error signal is delayed by one sample we get with \(h[k] = \delta[k-1]\)

\[\Phi_{e_H e_H}(\mathrm{e}^{\,\mathrm{j}\,\Omega}) = \Phi_{ee}(\mathrm{e}^{\,\mathrm{j}\,\Omega}) \cdot \left| 1 - \mathrm{e}^{\,-\mathrm{j}\,\Omega} \right|^2\]

For linear uniform quantization \(\Phi_{ee}(\mathrm{e}^{\,\mathrm{j}\,\Omega}) = \sigma_e^2\) is constant. Hence, the spectral shaping constitutes a high-pass characteristic of first order. The following simulation evaluates a noise shaping quantizer of first order.

```
In [1]:
```

```
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import scipy.signal as sig
w = 8 # wordlength of the quantized signal
A = 1 # amplitude of input signal
N = 32768 # number of samples
def uniform_midtread_quantizer_w_ns(x, Q):
# limiter
x = np.copy(x)
idx = np.where(x <= -1)
x[idx] = -1
idx = np.where(x > 1 - Q)
x[idx] = 1 - Q
# linear uniform quantization with noise shaping
xQ = Q * np.floor(x/Q + 1/2)
e = xQ - x
xQ = xQ - np.concatenate(([0], e[0:-1]))
return xQ[1:]
# quantization step
Q = 1/(2**(w-1))
# compute input signal
x = np.random.uniform(size=N, low=-A, high=(A-Q))
# quantize signal
xQ = uniform_midtread_quantizer_w_ns(x, Q)
e = xQ - x[1:]
# estimate PSD of error signal
nf, Pee = sig.welch(e, nperseg=64)
# estimate SNR
SNR = 10*np.log10((np.var(x)/np.var(e)))
print('SNR = %f in dB' %SNR)
plt.figure(figsize=(10,5))
Om = nf*2*np.pi
plt.plot(Om, Pee*6/Q**2, label='simulated')
plt.plot(Om, np.abs(1 - np.exp(-1j*Om))**2, label='theory')
plt.plot(Om, np.ones(Om.shape), label='w/o noise shaping')
plt.title('Estimated PSD of quantization error')
plt.xlabel(r'$\Omega$')
plt.ylabel(r'$\hat{\Phi}_{e_H e_H}(e^{j \Omega}) / \sigma_e^2$')
plt.axis([0, np.pi, 0, 4.5]);
plt.legend(loc='upper left')
plt.grid()
```

```
SNR = 45.128560 in dB
```

**Exercise**

- The overall average SNR is lower than for the quantizer without noise shaping. Why?