Digital Communications - Experiment DCL-1: Sampling and Quantization (ADC+DAC)

Introduction to Analog-to-Digital Conversion (ADC) and Digital-to-Analog Conversion (DAC)

• The interface between analog and digital domains is achieved through:
  • Analog-to-Digital Converter (ADC): Converts analog signals to digital.
  • Digital-to-Analog Converter (DAC): Converts digital signals back to analog.
• ADC Discretization Tasks:
  • Sampling: Converts continuous-time signals to discrete-time signals.
  • Quantization: Converts continuous amplitude values to discrete amplitude levels (steps).
• Both sampling and quantization transform quasi-infinite floating-point values into integer numbers.

Lab Experiment Focus and Setup

• This lab focuses on ADC components.
• Wiring and setup may be pre-configured due to time constraints, allowing focus on measurements.
• Hardware Experiment:
  • Performed using real hardware.
  • Groups of 2-4 students work on a single workbench.
  • Hardware resources are limited, necessitating some experiments in software.
• Software Experiment (Matlab Simulink):
  • Utilized when hardware is unavailable.
  • Commonly used in industries and research.
  • Instructions:
    • Copy provided files to a work directory.
    • Set the Matlab include path.
    • Run “initdigitalcommunications.m” initialization script.
    • Run the GUI “digitalcommunicationsgui.m”.
    • Follow on-screen steps by clicking the lab exercise button.
    • Use variables defined in “initdigitalcommunications.m” instead of constant numbers (e.g., c.sampling_frequency).

Important Notes on Units

• Frequency:
  • Be mindful of units.
  • If frequency is required in radians/second ($\omega$), use $\omega = 2\pi f$, not $f$ [Hz].

Sampling in Time (ADC)

• Prepare an analog input signal using a sine wave generator.
• Verify the output shape, frequency, and amplitude using an oscilloscope.
• Agilent 33500B Generator:
  • Displayed amplitude is Û, where $U(t) = \hat{U} \cdot \cos(\omega t)$, not $V_{pp}$ (peak-to-peak).
  • Adjust output voltage to match the ADC's valid input range (Û = 2.5V for a voltage swing of -2.5V to +2.5V, totaling 5V).
  • Adjust frequency to stay within the baseband bandwidth (e.g., 1kHz, 2kHz, 3kHz).

Sample-and-Hold Block (S&H)

• If available, connect an S&H unit to the source (input E1, output PAM1).
• Clock setting on the S&H box should be set to 4 = external.
• Apply a sampling clock (TTL signal, square wave) with a frequency ${f_s} = 8kHz$.
• Use the generator “PNG02” with a clock output of 8kHz.
• Measure input and output shapes via oscilloscope, along with the sampling clock.
  • Determine the trigger signal and appropriate time base (____ seconds/unit  ____ seconds/total horizontally).
• Adjust the pulse width setting ($\tau$) and observe the output (1=3s, 2=6s, 3=30s, 4=60s, 5=120s).
• Spectrum Analysis:
  • Measure the spectrum using a spectrum analyzer or FFT function on the oscilloscope.
  • Identify the dominant input frequency and its proximity to DC.
  • Determine its frequency and adjustments needed for better FFT visualization.
• Audio Output:
  • Listen to the S&H block’s output using headphones or active loudspeakers.
  • Use a personal audio source (CD/MP3-player, laptop, etc.) or the PC’s sound output.

Signal Frequency and Shape Variation

• Increase the input signal frequency beyond half of the sample frequency (to 1⋅${f<em>s}$ or 2⋅${f</em>s}$) and back.
  • Observe the effect on the oscilloscope and listen to the output.
• Change the signal shape from sine wave to ramp (linear increase with time) while keeping the frequency below ${f_s}/2$.
  • Verify the shape and listen to the sampled output.

Real Audio Signals

• Use a real audio signal source like music or voice from a Walkman, MP3 player, smartphone, laptop, or the lab PC.
• Typical Bandwidth:
  • Telephony signal: As per the bandwidth diagram.
  • Music: Assume frequencies up to 20kHz.
• Oscilloscope observation:
  • Channel A = input, Channel B = output of the S&H block.
  • Use a stationary input signal (long sound sample or A-B-repeat).
• Power Spectrum:
  • Obtain the power spectrum of the input and output signals using a spectrum analyzer or FFT function.
  • Explain the observed effects.

Aliasing Effect and Nyquist-Shannon Criterion

• Sampling Theorem:
  • If $x(t)$ contains no frequencies higher than $B$ Hz, it is completely determined by samples spaced $T = 1/(2B)$ seconds apart.
  • Sufficient sample rate: ${f_s} = 2B$ samples/second or higher.
  • Bandlimit for perfect reconstruction: $B \le {f_s}/2$.

Anti-Aliasing Filter

• Employ a low-pass (LP=TP) filter (“Butterworth”) before the S&H circuit.
• Adjust the cutoff frequency to attenuate the input signal to at least -40 dB (voltage) or -20 dB (power) at ${f_s}/2$.
• Use the “Butterworth TP” box with filter order $n = 20$.
• Repeat the experiments with the active anti-aliasing filter.
• Explain the measurement observations and auditory experiences.

Quantization (ADC)

• Attach the quantizer (ADC) with at least 8 digital outputs and a clock, replacing the S&H unit.
• Set the resolution to the maximum number of bits, if possible.
• Determine the number of discrete steps in the output.
• Bit Clock:
  • Set ${f<em>{bit}} = 8 \cdot {f</em>s}$, so for ${f<em>s} = 8kHz$, ${f</em>{bit}} = 64kHz$.
  • Provide this clock to “Bitclock_in”.
• Word Clock:
  • Observe and verify 8kHz at the “word clock” output (not an input).
• Clock Phases:
  • Examine the output phases of the bit clock and word clock on an oscilloscope, triggering on the word clock.
  • Identify the start of the byte, the MSB, and the LSB using a sawtooth/ramp input waveform.
  • Explain why the input and sampled signal move on the oscilloscope.

Reconstruction (DAC)

• Connect the full-resolution ADC and DAC using a parallel cable between “digital out parallel” and “digital in parallel”.
  • When connected via parallel cable, the output “digital out seriell” has no signal.
• Verify the polarity of the clock and sample signals; determine ${f_s}$ and explain the frequency differences.
• Use sine wave and/or sawtooth input signals (not music).
• Ensure the DAC output resembles the ADC input; correct if necessary.

Spectrum Analysis of Discrete-Time Signals

• Sketch the spectrum $S(f)$ of the input cosine wave $s(t)$ (frequency $f_i$).
• Sketch the spectrum ${S_s}(f)$ of the discrete-time signal $s(k)$.

Sample-and-Hold Reconstruction

• Verify with an oscilloscope that the DAC output has the shape of a rect($t/T_s$) reconstruction (sample-and-hold).
• Determine the spectrum $O(f)$ of the output signal $o(t)$.

Sin(x)/x Reconstruction

• If available, apply a different reconstruction/interpolator function and measure the spectrum again.

Reconstruction by Low-Pass Filter

• Use the sample-and-hold output (rect) and attach a low-pass filter (Butterworth-TP, $n = 20$, ${f<em>c} = 4kHz$) before ${f</em>s}$ to reconstruct the signal.
• Observe the output, measure the spectrum, and explain the results.

Amplitude Quantization (ADC)

Effect of Exceeding Input Voltage Range

• Increase the input voltage beyond the ADC's tolerable range (“clipping”).
• Observe the signal with an oscilloscope and listen to the effect.
• Repeat with music as input.

Effect of Limited Quantization Steps

• Limit the quantizer resolution to 3 bits by disabling LSBs (D0..D4 down, D5..D7 up).
• Demonstrate the input-output characteristic diagram using a sawtooth waveform and the X-Y-diagram of the oscilloscope.

Probability Distribution Function (PDF)

• Sketch or measure the PDF of the input signal amplitude for:
  • Sawtooth
  • Sine wave
  • Gaussian noise
• Determine if the oscilloscope can assist in calculating the PDF or histogram.

PDF of Quantization Noise Amplitude

• Describe the typical quantization error signal in the time domain and measure it.
• Determine the maximum voltage and its value in dBmV.
• Explain the PDF distribution of its magnitude.
• Calculate the noise power level in mW and dBm for the 8-bit resolution ADC, assuming $R = 1\Omega$.

Spectral Distribution of Quantization Noise

• Sketch the spectrum of the quantization noise.
• Measure the spectrum.

Oversampling and Digital Low-Pass Filtering

• Examine what happens when the sampling frequency is doubled (from ${f<em>s}$ to 2${f</em>s}$).
• Determine if the SNR changes.
• Low-pass filter the discrete signal $s(k)$ to a cut-off frequency of 2${f_s}/4$ and analyze how the SNR changes and explain why.

Measuring Distortion and Signal-to-Noise Ratio (SNR)

• Design a circuit to measure the SNR of the quantized signal.
• Identify appropriate tools for this task.
• Supply a sine wave input of maximum tolerable amplitude.
• Measure how the DAC output changes with different quantization steps (1 to 8 bits).
• Verify measurements against the theoretical curve.
• Use the “Klirrfaktor-Analyser” and matched band-pass/band-stop filters.
• Note that the 0dB output may provide a -40dB (amplitude 1/100) version of the distortion.