# Advanced Digital Signal Processing - Sampling and Quantization

### 1. Relationship between continuous and discrete signals

A complex-valued continuous-time signal $$v_a(t)$$ has the Fourier transform shown in the figure 1. Figure 1: Fourier transform of $$v_a(t)$$

This signal is sampled to produce the sequence  $$v(n)=v_a(nT)$$.

1. Sketch the Fourier transform $$V(e^{j\Omega})$$ of the sequence $$v(n)$$ for $$T=\frac{\pi}{\omega_2}$$.
2. What is the lowest sampling frequency that can be used without incurring any aliasing distortion, i.e. so that $$v_a(t)$$ can be recovered from  $$v(n)$$?

## Amount and difficulty

• Working time: approx. 20 minutes
• Difficulty: easy

### 2. Overall system for filtering a continuous-time signal in digital domain

Figure 2 shows an overall system for filtering a continuous-time signal using a discrete-time filter. The frequency response of the ideal reconstruction filter $$H_\textrm{r}(j\omega)$$ and the discrete-time filter are shown below. Figure 2: Overall system

1. For $$V_\textrm{a}(j\omega)$$ as shown in figure 3 and $$1/T=20kHz$$ sketch $$V_\textrm{i}(j\omega)$$ and $$V(e^{j\Omega})$$. Figure 3: Spectrum of $$V_\textrm{a}(j\omega)$$ and $$H_\textrm{eff}(j\omega)$$

For a certain range of values of T, the overall system, with input $$v_\textrm{a}(t)$$ and output $$y_\textrm{r}(t)$$, is equivalent to a continuous-time lowpass filter with frequency response $$H_\textrm{eff}(j\omega)$$ sketched in figure 3.

1. Determine the range of values of $$T$$ for which the information presented above is true, when $$V_\textrm{a}(j\omega)$$ is
bandlimited to $$|\omega | \leq 2\pi \cdot 10^4$$ as shown in figure 3.
2. For the range of values determined in (b), sketch $$\omega_c$$ as a function of $$1/T$$.

Note: This is one way of implementing a variable-cutoff continuous-time filter using fixed continuous-time and discrete-time filters and a variable sampling rate.

## Amount and difficulty

• Working time: approx. 30 minutes
• Difficulty: easy

### 3. Quantization

A sinusoid signal $$v(n) = 5 \sin (\frac{\omega_0}{\omega_{\textrm{s}}} \cdot n)$$ with $$f_0=5\textrm{ Hz and } f_{\textrm{s}}=10\textrm{ kHz}$$ has to be quantized $$\textrm{(}v_{\textrm{q}} = Q[v(n)]\textrm{)}$$ with a midtreat quantizer. The range of the signal is $$\pm 5$$ V and the word length of the quantizer 4 bits. The quantizer at digital full scale.

1. How many quantization levels $$L$$ does the quantizer have? What is the value of $$\Delta$$?
2. Sketch the input-output characteristic of the quantizer. How different is a midtreat quantizer to a midrise quantizer.
3. For time index $$n = 1250$$ calculate the quantized value $$v_{\textrm{q}}(n)$$, the quantization error $$e_{\textrm{q}}(n)$$ and represent $$v_{\textrm{q}}(n)$$ using bipolar code (sign and magnitude representation).

The quantization error over time can be modeled as a noise that is added to the input signal.

1. Sketch the real system and the mathematical model of the system with the added quantization noise.
2. Calculate the power $$P_{\textrm{n}}$$ of the quantization noise.

Determine the SNR in dB and in The signal's amplitude is changed to $$\pm 1$$ V, while the range $$R$$ of the quantizer remains

1. How is SNR affected with this change?
2. What word length has to be chosen to achieve an SNR > 45 dB?

## Amount and difficulty

• Working time: approx. 20 minutes
• Difficulty: middle