The Bridge from Analog to Digital

Sampling is the first crucial step in the process of converting an analog signal into a digital one. It transforms a signal that is continuous in time into a signal that is discrete in time. Imagine watching a car drive down a road; sampling is like taking a series of snapshots of the car at perfectly regular time intervals. Each snapshot captures the car's position at that specific moment, creating a sequence of images that represents its journey.

In telecommunications, instead of a car, we are capturing the value (amplitude) of an analog signal, like a sound wave, at these discrete moments. The result is a series of measurements called samples.

The Golden Rule: The Nyquist-Shannon Sampling Theorem

A critical question arises: how often do we need to take these "snapshots" to be able to perfectly reconstruct the original analog signal? If we sample too slowly, we might miss important changes. The answer lies in one of the most important principles of digital signal processing: the Nyquist-Shannon Sampling Theorem.

$f_s \ge 2 \cdot f_{\text{max}}$

• Sampling Frequency ($f_s$): This tells us how many samples are taken per second.
• Maximum Signal Frequency ($f_{\text{max}}$): This is the highest frequency contained within the signal we want to digitize.

Intuitively, this means we need to take at least two samples for every cycle of the highest frequency wave in our signal-one to capture the peak and one to capture the trough-to be able to reconstruct its shape.

Practical Example: Digitizing Human Voice for Telephony "

The Nyquist-Shannon theorem is not just theoretical; it defines the fundamental parameters of digital telephony.

1. Determine $f_{\text{max}}$: For intelligible speech, the essential frequencies lie between 300 Hz and 3400 Hz. Therefore, we set $f_{\text{max}} = 3400 \text{ Hz}$. For practical purposes and filter design, this is often considered to be within a 4 kHz band.
2. Calculate Minimum Sampling Rate: According to the theorem, the minimum sampling frequency must be $f_s \ge 2 \cdot 3400 \text{ Hz} = 6800 \text{ Hz}$.
3. Set the Industry Standard: To provide a margin of safety and simplify filter design, the telecommunications industry standardized the sampling frequency for voice at 8000 Hz (8 kHz). This value is fundamental to digital telephony.

A direct consequence of this standard is the sampling period ($T_s$), the time between consecutive samples:

This 125 microsecond interval is the basic time frame in many digital communication systems, including PCM and SDH/SONET.

The Pitfall of Undersampling: Aliasing

What happens if we violate the Nyquist-Shannon theorem and sample too slowly ($f_s < 2 \cdot f_{\text{max}}$)? The result is an irreversible error known as aliasing.

A classic visual analogy is the "wagon-wheel effect" in films. When a spoked wheel rotates quickly, if the camera's frame rate (its sampling rate) is too low, the wheel can appear to be spinning slowly backwards or even standing still. The high frequency of the wheel's rotation has been aliased into a false, lower frequency. The same thing happens with signals: a high-frequency sine wave, if undersampled, will appear in the digital version as a completely different, lower-frequency sine wave.

The Solution: The Anti-Aliasing Filter

To prevent aliasing, we must ensure that no frequencies above $f_s / 2$ ever enter the sampler. This is achieved by placing a low-pass (anti-aliasing) filter before the sampling circuit. For telephony, this filter cuts off all frequencies above approximately 4 kHz before the 8 kHz sampler.

The Output of Sampling: The PAM Signal

The direct result of the sampling process is a new type of signal called a Pulse Amplitude Modulation (PAM) signal. In a PAM signal, the amplitude of a train of regularly spaced pulses is varied in proportion to the value of the analog signal at the sampling instants.

This signal is discrete in time (it only has values at the sampling moments), but it is still analog in its values, as the pulse amplitudes can take on any value from the original signal's range. This signal is generated by a circuit called a Sample-and-Hold (S&H). The PAM signal is the bridge to the next step in digitization: Quantization.