Sampling & Aliasing

Another fundamental mathematical theorem in communication engineering is Nyquist’s Sampling Theorem. Nyquist’s Theorem states that a signal containing frequency components up to a maximum frequency of B Hz (in other words the signal is bandlimited to B Hz), can be perfectly reconstructed if it is sampled at or greater than 2B samples per second. Therefore, for an audio signal with a maximum frequency component at 20kHz, this signal would need to be sampled with a sampling frequency of at least 40kHz. In information theory language, all the information in the signal is then contained in the sample values and so the original signal can be perfectly recreated. 

If you fail to sample at least as high as this frequency of twice the highest frequency component then the samples will contain aliasing products in the spectrum. Aliasing refers to the fact that the sampled values of a sinusoid whose frequency is below the Nyquist frequency (half the sampling frequency) are exactly the same sample values as would be found if a sinusoid with certain other frequencies was sampled. That is, these other sinusoids are aliases of the sinusoid whose frequency is below the Nyquist frequency, hence the term. The following holds: 

“When sampling at a rate of F_s samples/s, if k is any positive or negative integer, we cannot distinguish between the sampled values of a sinewave of f_0 Hz and a sinewave of (f_0 + k * f_0) Hz.” 

[Understanding Digital Signal Processing, R. G. Lyons, Addison Wesley, 1997] 

Therefore, it is extremely important not to allow frequency components with any significant power at frequencies above the Nyquist frequency to be present in the sampled signal. To achieve this situation an analogue (it has to be analogue) anti-aliasing filter is used to bandlimit the signal prior to sampling.

Sampling consists of a “sample and hold” circuit followed by conversion a digital code word, and is implemented using a chip called an ADC (analogue to digital converter). For an audio signal, the output of the ADC will be a code word which is a binary word whose number in binary represents the amplitude of the sampled signal. A typical number of bits for an ADC that is used to sample an audio signal will be 16 or 24 bits. This means there will be 2^16 or 2^24 possible quantisation levels respectively. An ADC used for audio would use linear quantisation which means that all the quantisation levels are equally spaced. Because the digital sample values are discrete there will be a small error between the actual analogue voltage and the encoded amplitude. This error is called quantisation noise. Increasing the number of bits of the ADC decreases the voltage difference between adjacent levels (increases the resolution) and therefore the quantisation noise is reduced. The maximum value of the quantisation error is equal to half the voltage of the least significant bit. For a 16-bit ADC this value is only 2^-17 = 0.0008% of the full-scale voltage. 

To reduce this even further you can use a dither signal which is a higher frequency random signal that is added to the signal prior to sampling. This signal modulates the value of the least significant bit and produces a pulse width modulation signal, where the duty cycle of the pulse (percentage of the time the pulse is high) is proportional to the value of the voltage level above the lowest quantisation levels divided by the resolution (i.e. the smallest quantisation step, or the voltage difference between 2 levels) of the ADC. When the signal is later reconstructed using a lowpass filter after the digital to analogue converter (DAC), the filter integrates (adds the contributions of) this pulse width modulation signal to reproduce values closer to the original analogue sample values. This effectively increases the resolution of the ADC and reduces the quantisation noise to below that which would result if the dither signal was not added, i.e. the quantisation error is reduced to below half the least significant bit and so the accuracy of the ADC is increased. 

The job of the ADC is to store the amplitude value of the analogue signal as a digital number which is proportional to the amplitude of the analogue signal voltage. This is called pulse code modulation. As these numbers are in binary format the number of bits defines how many discrete levels can be used. For an N-bit PCM codeword there are 2^N different codewords, for example, if an 8-bit codeword was used then there are 2^8 = 256 discrete amplitudes and therefore discrete codewords that can be used.

A different type of ADC to this is the oversampling DAC. This samples at a far faster rate than that used for the above type of ADC and just uses the rule that if the signal is higher than the present sampled value (which is fed back to an analogue comparator) then a digital 1 is output, otherwise a digital 0 is output. In this way the ADC follows the amplitude of the audio waveform because the ADC’s quantised step combined with the far faster sampling frequency allows the fed back analogue value (produced by converting the quantised value back to analogue using a DAC) to move faster than the audio signal is able to. One of the benefits of the oversampling ADC is that the noise spectrum is shaped so that the noise power is moved up in frequency and the noise power in the band of interest, i.e. the audio band, is reduced in power. The spectrum of the noise actually resembles an upside down cosine wave with the noise starting at a lower power at DC and increasing in power as the frequency increases, so the majority of the quantisation noise power that would have resulted if a normal ADC had been used would be moved out of the audio band and therefore the signal to quantisation noise power ratio (SNqR) is increased.

DAC & Signal Reconstruction

To reconstruct the analogue signal that was sampled by the ADC and encoded as a sequence of PCM numbers, a digital to analogue converter (DAC) is used in conjunction with a lowpass filter. The DAC does the opposite to that which the ADC does. That is, it takes a PCM encoded number (for example a 16- or 24-bit number) and outputs an analogue voltage that is proportional to the value that the PCM number signifies. This analogue voltage is then held constant by a sample-and-hold circuit so the analogue waveform consists of a sequence of rectangular pulses all next to each other and with different amplitudes. To finally reconstruct the original signal these rectangular pulses are sent through a lowpass filter. This smooths the waveform to produce exactly the original signal assuming a perfect lowpass filter. Theoretically, the sampling of the original signal, followed by reconstruction using a DAC and lowpass filter will perfectly reconstruct the original signal. There will in practice be small errors because it is impossible to construct perfect ‘brick-wall’ filters but it is possible to obtain a reconstructed signal with a very small error.