# Revisiting Nyquist

The 20th century brought with it the need for the transport of pictures and sound. A large number of engineers and scientists working for Bell Telephone Laboratories became engaged in studies leading to solutions of imminent as well as long-term needs.

One of these talented people was the prolific Harry Nyquist. Among his earliest work was the development of the frequency division multiplexing (FDM) concept. FDM used individual and harmonically related (multiples of 8kHz) carriers, each of them amplitude-modulated with a specific message: telephone quality audio with a frequency range of 300Hz to 3.4kHz.

Early implementations used double-sideband AM. Later approaches used single-sideband suppressed-carrier concepts. This allowed the doubling of the number of messages by assigning different messages to the lower and upper sideband. The number of the individual carriers used depended on the cable losses.

The invention of the electronic tube by Lee De Forest introduced the use of amplifiers to compensate for cable losses. But amplifiers have shortcomings and introduce linear and nonlinear distortions, which can lead to mutual interference between the multiplexed messages. Nyquist worked on reducing these defects by developing and applying the negative feedback concept.

While FDM allows for the transmission of multiple simultaneous messages, a better approach — time division multiplexing (TDM) — gives superior results. In TDM, several different signals are transmitted sequentially over a single channel. Nyquist became interested in and developed the TDM concepts with early application to telephone circuits. The signals transmitted by TDM are pulse code modulated (PCM). The basis of PCM is Nyquist's sampling theorem. According to the sampling theorem, we can convey the entire message of an information signal by sending only the amplitudes of the signal at specific instants, called sampling times.

The sampling concept

The sampling of an analog signal consists of checking its amplitude at regular intervals (T). Nyquist's sampling theorem stipulates that the sampling frequency, Fs = 1/T, be at least twice the maximum baseband frequency Fs>2Fb.

Figure 1 shows the sampling mechanism of a sinewave. The amplitude of the sampled sinewave is measured at constant time intervals, T. The amplitude of the samples is modulated by the sampled frequency, resulting in a process of pulse amplitude modulation (PAM).

Figure 2 shows an example of a sinewave F sampled at twice its frequency. It is intuitively evident that there are sufficient samples to allow for the reconstruction of the original waveform.

Figure 3 shows an ideal spectrum of a PAM process, where Fs = 2Fb and Fb is the baseband spectrum. The PAM spectrum resembles an AM spectrum, except that in addition to the Fs carrier and its sidebands, there are spectral components at multiples of Fs. In the real world, the baseband spectrum exceeds ½Fs. To avoid the generation of spurious responses known as “aliasing,” the base-band spectrum has to be limited to less than ½Fs through the use of a well-designed brick-wall low-pass filter. A good low-pass filter features a sharp cutoff while avoiding passband ripple effects and high-frequency group delays, which degrade the analog signal characteristics and result in an unacceptable performance.

Figure 4 shows an example of a sinewave F sampled at 1.33 times its frequency. The result is an insufficient number of samples, and the original waveform cannot be reconstructed. The dotted line represents the reconstructed sinewave with a frequency of F/3.

Figure 5 shows the PAM spectrum, where Fs<2Fb. As a result, the lower sideband of Fs overlaps the baseband Fb, resulting in aliasing. The aliasing components of the PAM spectrum result in audible or visible (as the case may be) spurious low-frequency spectral components, which cannot be eliminated.

Figure 6 on page 34 shows the PAM spectrum of a sampled filtered baseband signal. Note that Fs>2Fb to allow the design of realizable and cost-effective low-pass filters with minimum ripple and high-frequency group delay.

PAM results in a sequence of pulses whose amplitude is proportional to the amplitude of the sampled analog signal at the sampling instant. The process of PCM helps represent the amplitudes of the successive samples of the analog waveform by binary numbers. Thus, an infinite number of possible pulse amplitude values are converted to a finite number of discrete levels Q according to the expression Q = 2n, where n is the number of bits per sample. Telephone audio (voice) is sampled with a resolution of 8 bits per sample. Video signals are sampled with a resolution of 8 bits or 10 bits per sample. Audio signals are sampled with a resolution of 16 bits per sample (CD format) and between 20 and 24 in studio productions. Once in binary form, the numbers are transmitted as on (1) and off (0) pulses. Each sequence of pulses is a code for a sample amplitude, hence the name of pulse code modulation. The capacity of a PCM signal (bit rate) is expressed in bits-per-second (bps) obtained by multiplying the number of bits per sample (n) by the sampling frequency (Hz). The required transmission bandwidth is related to the bit rate.

Common carrier sampling hierarchy

Common carriers have developed a set of digital transmission hierarchies, which are a multiple of 64kb/s, the bit rate required to send a voice signal (bandwidth 300Hz to 3400Hz, sampled at 8kHz with an accuracy of 8 bits per sample) for telephone conversations. Table 1 lists some of the more common digital distribution hierarchies used in North America.

Transmitting digital SMPTE 259M bit-serial component digital signals with a bit rate of 270Mb/s requires either the displacement of 4032 telephone conversations (DS4) or some means of compression to fit into one of the other available channels. One of the most popular digital hierarchies in North America is DS3, colloquially referred to as 45Mb/s.

DS3 was initially used to transmit composite NTSC signals sampled at 10.7MHz with an accuracy statistically varying between 6 bits per sample and 9 bits per sample, depending on the picture complexity. The compression used was known as differential pulse code modulation (DPCM), and it is obsolete today. The equivalent hierarchy level in Europe offers a bit rate of 33.368Mb/s and was used to transmit DPCM composite PAL signals.

As the state of the art improved, MPEG-2 signals with a bit rate of the order of 8Mb/s offer a picture quality subjectively superior to that of a 45Mb/s DPCM composite NTSC signal. A DS3 45Mb/s channel can accommodate several multiplexed SDTV MPEG-2 8Mb/s digital signals of distribution quality and this, in the long run, will contribute to the demise of the obsolete and spectrum-wasting DPCM compression schemes. With the advent of DTV with a terrestrial transmission bit rate of the order of 19.4Mb/s, an entirely new set of signal distribution scenarios have yet to unfold.

Michael Robin, a fellow of the SMPTE and former engineer with the Canadian Broadcasting Corp.'s engineering headquarters, is an independent broadcast consultant located in Montreal, Canada. He is co-author of Digital Television Fundamentals, published by McGraw-Hill, and recently translated into Chinese and Japanese.