AET 2050 - Daw Production

3. Conversion Principles - Sampling

Objectives:

1. Identify the two processes involved in converting an analog signal into a digital representation.
2. Define the Sampling Theorem and identify those responsible for developing it.
3. For a given sampling frequency, calculate the Nyquist frequency and the expected frequency response.
4. Determine the sampling frequency needed for a given signal bandwidth.
5. Define aliasing, and describe its effects on reconstructed audio.

Chapter 2, Desktop Audio Technology - Rumsey

The power of the computer allows us to process and manipulate data in virtually any way desired; however, for audio to be processed, the digital representations of the audio must be created, and after such processing, the data must be returned to analog audio again.

This unit explores the process of converting analog audio signals into digital representations, and converting them back again.

# The Conversion Process

Analog-to-digital conversion (A/D) is the process of converting analog signals into digital representations. It involves creating a set of discreet values that correspond to the original signal.

Amplitude measurements are taken at specific time intervals, and those measurements are converted into digital data. The process is like placing the signal onto a grid. The x-axis is time and the y-axis is amplitude.

The process of taking measurements at specific time intervals is called sampling, and the process of assigning a value to those measurements is called quantization.

Digital-to-analog conversion (D/A) is the process of reconstructing the analog signal from the digital data. The data is transformed into a series of pulses, which are joined together to reform the original audio.

# Sampling

The analog audio signal is a time-continuous waveform. The waveform is measured, or sampled, at periodic intervals.

These samples form a series of time-discreet amplitude pulses that, together and in sequence, form a representation of the continuous waveform. ## The Sampling Theorem

There must be enough samples to faithfully reproduce the frequency in question. We could take as many samples as possible, but there are reasons to limit the number:

• Hardware limitations - hardware can only gather the samples so fast.

• Storage limitations - the samples must be stored, and there is only so much room available to store the given samples.

How many samples are needed? There must be enough to accurately reproduce any and all frequencies in the original signal.

 A Brief History of Sampling Research 1915 - E.T. Whitaker devised a proof showing that a band-limited function can be reconstructed from samples. 1920 - K. Ogura proved that if a function is sampled at a frequency at least twice the highest function frequency, it could be reconstructed from those samples. 1928 - Bell Labs engineer Harry Nyquist published an article titled Certain topics in Telegraph Transmission Theory.In this article he provided proof that for complete signal construction, the frequency bandwidth is proportional to the signaling speed, and that the highest frequency is equal to half the number of code elements per second. 1949 - Claude Shannon unified many aspects of sampling, founded that larger science of information theory.

The theorem that defines the sampling process is generally attributed to Harry Nyquist, and is known as the Nyqist Theorem.

The Nyquist theorem states that:

• A continuous band-limited signal can be replaced by a discreet sequence of samples without loss of information, and the original signal can be reconstructed from those samples.

• The sampling frequency must be at least twice the highest signal frequency.

More specifically, audio signals with frequencies between 0 and fS/2 Hz can be exactly represented by S samples per second.

If there are at least two samples for a given frequency, that frequency can then be reproduced, since there is a sample for the positive and the negative portion of each cycle. The reproduction process can then recreate both sides of the cycle and, therefore, the reconstruct the given frequency.

The highest frequency fS/2 that can be reproduced for a given sampling frequency S is known as the Nyquist frequency

If a frequency is sampled at less than two times per cycle, a different frequency will be reconstructed.

## Sampling Process

The continuous signal must be measured or "sampled" at regular time intervals. Samples are like continuous "still frames" of the signal.

A pulse train, at the sampling frequency, is modulated by, or mixed with, the audio signal. The result is a pulse train with the amplitude of the pulses being modified to match the original signal. This process is known as Pulse Amplitude Modulation (PAM). Amplitude modulation produces sidebands above and below the sampling frequency, as well as all harmonics of the sampling frequency. (The sampling pulse contains many overtones.) The sidebands are pairs of "mirror images" of the original audio signal.

## Aliasing and Filtering

If frequencies above the Nyquist frequency are sampled, the result is an overlapping of the images. These overlapped frequencies result in aliasing - unwanted frequencies from the higher images producing errant frequencies in the signal. Results of undersampling.  A new alias frequency is created. Aliasing. a) audio limited below fN creates an image above fN. b) Audio extends beyond Nyquist frequency, creating aliased audio below fN.   c) 1kHz tone sampled at 30 kHz, producing sidebands +- 1kHz from images.  d) 17kHz tone sampled at 30 kHz, producing sidebands +- 17kHz from images.

To avoid aliasing, an anti-aliasing filter is used to block out all frequencies above the Nyquist frequency. Real-world filters cannot perfectly filter out frequencies at the exact edge of the desired program spectrum; therefore, Nyquist frequencies are set higher than the top program frequency to account for filter rolloff.

# Reconstruction

The pulses can be passed through a low-pass filter and reproduce the original audio.

In the frequency domain, the upper sidebands are eliminated, leaving the original audio spectrum. In the time domain, the pulses are filtered, resulting in an impulse response.

When joined together, the impulses sum to reconstruct the original signal. When considered in the frequency domain, the filter is called an anti-imaging filter.

When considered in the time domain, the filter is called a reconstruction filter.

Both are accurate descriptors of its function.

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