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