**A**merican physicist, electrical and communications engineer, a prolific inventor who made fundamental theoretical and practical contributions to telecommunications. The Sweden years

Harry Nyquist’s parents Lars Jonsson and Katarina Eriksdotter got married 1879. The year after they bought a farm in Tomthult together with Olof Jonsson a brother to Lars. The farm is called “Dar Sor” and is situated 40 kilometers north of Karlstad, the main town in this region called “Varmland”. In 1894 the couple released Olof from the farm. An interesting fact is that the family was baptists when the Swedish church is Lutheranian.

The name Jonsson had to be changed because just hundred meters away there lived another Lars Jonsson and there was huge problem with themail delivery. Therefore they agreed to change names, which not was a rare thing to do at this time. Harry’s father changed the name to Nyquist. Harry was the fourth child of eight and was born on 7 February 1889 in Nilsby, Sweden.

The other children were Elin, Astrid, Selma, Ameli, Olga, Axel and Berta. The family was far from rich, but still the children were allowed to study six years in school and after that the continuing school with more concentrated education. Harry went to three different school houses. This because the old school burnt down in 1899 and during the building of a new school the education was held in Nilsby Mission-Hall.

Harry went also to school in the new building that was finished in 1900. Parallelly Harry helped his father in his shoemaker’s shop and the farm. Harry’s teacher Moden put a lot of confidence in Harry and Harry could even borrow books from his teacher (not common in those days). Moden wanted Harry to be a teacher. When Harry pointed out that his family was poor the teacher suggested that he should emigrate to America because the chances were bigger there. Two of Moden’s sons had already done that.

Harry was at that time 14 years old. The following years he worked at the construction of the sulfate factory in Deje in order to fulfill the demands on emigration and to get travel money: 10 dollars and a guarantee that he has a job in America. It took 4 years of hard work to fulfill the goal – to emigrate to America.

Education and Career in the U.S.A.

Nyquist moved to the United States in 1907. Harry Nyquist came to the University of North Dakota, Grand Forks, in 1912, where he earned his Bachelor of Science in Electrical Engineering degree in 1914 and his master of Science in Electrical Engineering degree in 1915. Nyquist continued his graduate studies at Yale University, New Haven, Conn., where he received the Ph.D. in physics in 1917.

He was employed at American Telephone and Telegraph Company (AT&T) from 1917 to 1934, in the Department of Development and Research Transmission, where he was concerned with studies on telegraph picture and voice transmission.

From 1934 to 1954 he was with the Bell Telephone Laboratories, Inc., where he continued in the work of communications engineering, especially in transmission engineering and systems engineering. At the time of his retirement from Bell Telephone Laboratories in 1954, Nyquist was Assistant Director of Systems Studies.

During his 37 years of service with the Bell System, he received 138 U.S. patents and published twelve technical articles. His many important contributions to the radio art include the first quantitative explanation of thermal noise, signal transmission studies which laid the foundation for modern information theory and data transmission, the invention of the vestigial sideband transmission system now widely-used in television broadcasting, and the well-known Nyquist diagram for determining the stability of feedback systems.

Harry Nyquist (right) with John R. Pierce (left) and Rudolf Kompfner (center) – all scientists working for Bell Labs (1960). Bell Labs scientist John R. Pierce uncovered the principles and mathematics for the stable operation of the traveling wave tube, a device that amplifies microwave frequencies at very high power. The technology is used in space – vehicles and satellite guidance systems, and in communications networks.

Some of Nyquist’s best-known work was done in the 1920s and was inspired by telegraph communication problems of the time. Because of the elegance and generality of his writings, much of it continues to be cited and used. In 1924 he published “Certain Factors Affecting Telegraph Speed,” an analysis of the relationship between the speed of a telegraph system and the number of signal values used by the system.

His 1928 paper “Certain Topics in Telegraph Transmission Theory” refined his earlier results and established the principles of sampling continuous signals to convert them to digital signals. The Nyquist sampling theorem showed that the sampling rate must be at least twice the highest frequency present in the sample in order to reconstruct the original signal.

These two papers by Nyquist, along with one by R.V.L. Hartley, are cited in the first paragraph of Claude Shannon’s classic essay “The Mathematical Theory of Communication” (1948), where their seminal role in the development of information theory is acknowledged.

In 1927 Nyquist provided a mathematical explanation of the unexpectedly strong thermal noise studied by J.B. Johnson. The understanding of noise is of critical importance for communications systems. Thermal noise is sometimes called Johnson noise or Nyquist noise because of their pioneering work in this field.

In 1932 Nyquist discovered how to determine when negative feedback amplifiers are stable. His criterion, generally called the Nyquist stability theorem, is of great practical importance. During World War II it helped control artillery employing electromechanical feedback systems.

His remarkable career included advances in the improvement of long-distance telephone circuits, picture transmission systems, and television. Dr. Nyquist’s professional, technical, and scientific accomplishments are recognized worldwide.

It has been claimed that Dr. Nyquist and Dr. Claude Shannon, another signal procession pioneer, are responsible for virtually all the theoretical advances in modern telecommunications. He was credited with nearly 150 patents during his 37-year career.

His accomplishments underscore the excellent preparation in engineering that he received at the University of North Dakota. In addition to Nyquist’s theoretical work, he was a prolific inventor and is credited with 138 patents relating to telecommunications.

**Nyquist and FAX**

In 1918 H. Nyquist began investigating ways to adapt telephone circuits for picture transmission. By 1924 this research bore fruit in “telephotography” – AT&T’s fax machine. The principless used in 1924 were the same as those used today, though the technology was comparatively crude. A photographic transparency was mounted on a spinning drum and scanned. This data, transformed into electrical signals that were proportional in intensity to the shades and tones of the image, were transmitted over phone lines and deposited onto a similarly spinning sheet of photographic negative film, which was then developed in a darkroom.

The first fax images were 5×7 photographs sent to Manhattan from Cleveland and took seven minutes each to transmit.

In the late 1920s, the only technology to preserve musical recordings was to copy sound waves in wax. Harry Nyquist, an AT&T scientist, thought there was a better way. He wrote a landmark paper (Nyquist, Harry, “Certain topics in Telegraph Transmission Theory,” published in 1928) describing the criteria for what we know today as sampled data systems.

Nyquist taught us that for periodic functions, if you sampled at a rate that was at least twice as fast as the signal of interest, then no information (data) would be lost upon reconstruction. And since Fourier had already shown that all alternating signals are made up of nothing more than a sum of harmonically related sine and cosine waves, then audio signals are periodic functions and can be sampled without lost of information following Nyquist’s instructions. This became known as the Nyquist frequency, which is the highest frequency that may be accuratelysampled, and is one-half of the sampling frequency.

Harry Nyquist (1920′s) showed that to distinguish unambiguously between all signal frequency components we must sample at least twice the frequency of the highest frequency component, Figure 1.

Figure 1: In the diagram, the high frequency signal is sampled twice every cycle. If we draw a smooth connecting line between the samples, the resulting curve looks like the original signal. This avoids aliasing. The highest signal frequency allowed for a given sample rate is called the Nyquist frequency.

Harry Nyquist thought of a way to take an analog signal (such as voice) and code it (just like with the Morse code) using ones (1) and zeros (0). For this, he invented something called a “CODEC” or coder-decoder. This thing that today is the size of a fingernail (a microchip) measures the input analog signal, codes the result of the measurement and sends this code down the telephone lines and trunks. It does so often enough so its peer at the other end of the line can reconstruct the voice signal almost as good as it was at the calling side. N. Erd calls the measuring of the signal “sampling.” Good old Harry Nyquist also recommended that the number of samples per second for a good representation of the signal has to be twice as big as the number of Hertz of the fastest sine wave contained in the analog signal. Since the telephone only allows 4 kHz through the phone line, sampling for voice is done 8000 times per second.

Signal Sampling Theory was an exercise in frustration for Nyquist, since it needed 30,000 samples a second to make it work, and no system at that time could measure, record, store and reread that much information that quickly. He had to wait for computers, binary language, transistors and integrated circuits – 60 years of technological progress – to make digital recording and playback a reality.

The sampling theorem states that for a limited bandwidth (band-limited) signal with maximum frequency fmax, the equally spaced sampling frequency fs must be greater than twice of the maximum frequency fmax, i.e., fs > 2 fmax

in order to have the signal be uniquely reconstructed without aliasing. The frequency 2 fmax is called the Nyquist sampling rate. Half of this value, fmax, is sometimes called the Nyquist frequency. The sampling theorem is considered to have been articulated by Nyquist in 1928 and mathematically proven by Shannon in 1949. Some books use the term “Nyquist Sampling Theorem”, and others use “Shannon Sampling Theorem”. They are in fact the same sampling theorem.

The sampling theorem clearly states what the sampling rate should be for a given range of frequencies. In practice, however, the range of frequencies needed to faithfully record an analog signal is not always known beforehand. Nevertheless, engineers often can define the frequency range of interest. As a result, analog filters are sometimes used to remove frequency components outside the frequency range of interest before the signal is sampled.

For example, the human ear can detect sound across the frequency range of 20 Hz to 20 kHz. According to the sampling theorem, one should sample sound signals at least at 40 kHz in order for the reconstructed sound signal to be acceptable to the human ear. Components higher than 20 kHz cannot be detected, but they can still pollute the sampled signal through aliasing. Therefore, frequency components above 20 kHz are removed from the sound signal before sampling by a band-pass or low-pass analog filter. Practically speaking, the sampling rate is typically set at 44 kHz (rather than 40 kHz) in order to avoid signal contamination from the filter rolloff.

What if an engineer is interested in sampling a mechanical signal across ALL frequencies? Most mechanical signals have frequencies limited to below 100 kHz. Therefore, using a 200 kHz sampling rate should satisfy most mechanical engineering applications. The price for such a high sampling rate will be the huge amount of sample data to be stored and processed. Note that this limit should NOT be applied to electric engineering, where signals can contain much higher frequencies!

Figure 2: Graphically, if the sampling rate is sufficiently high, i.e., greater than the Nyquist rate, there will be no overlapped frequency components in the frequency domain. A “cleaner” signal can be obtained to reconstruct the original signal. This argument is shown graphically in the frequency-domain schematic.

Nyquist Plot of Impedance Spectra

The expression for Z(w) is composed of a real and an imaginary part. If the real part is plotted on the X-axis and the imaginary part on the Y-axis of a chart, we get a “Nyquist plot” (Figure 3). Notice that in this plot the Y-axis is negative and that each point on the Nyquist plot is the impedance at one frequency.

Figure 3 has been annotated to show that low frequency data are on the right side of the plot and higher frequencies are on the left. This is true for Electrochemical Impedance Spectra (EIS) data where impedance usually falls as frequency rises (this is not true of all circuits). On the Nyquist plot the impedance can be represented as a vector of length |Z|.

The angle between this vector and the x-axis is f , where f = arg(Z). Nyquist plots have one major shortcoming. When you look at any data point on the plot, you cannot tell what frequency was used to record that point. The Nyquist plot in Figure 3 results from the electrical circuit of Figure 4. The semicircle is characteristic of a single “time constant”. Electrochemical Impedance plots often contain several time constants. Often only a portion of one or more of their semicircles is seen.

A metal covered with an undamaged coating generally has a very high impedance. The equivalent circuit for such a situation is in Figure 6. The model includes a resistor (due primarily to the electrolyte) and the coating capacitance in series. A Nyquist plot for this model is shown in Figure 5.

The value of the capacitance cannot be determined from this Nyquist plot. It can be determined by a curve fit or from an examination of the data points. Notice that the intercept of the curve with the real axis gives an estimate of the solution resistance. The highest impedance on this graph is close to 1010 W. This is close to the limit of measurement of most EIS systems.

The Randles cell is one of the simplest and most common cell models. It includes a solution resistance, a double layer capacitor and a charge transfer or polarization resistance. In addition to being a useful model in its own right, the Randles cell model is often the starting point for other more complex models. The equivalent circuit for the Randles cell is shown in Figure 8. The double layer capacity is in parallel with the impedance due to the charge transfer reaction.

The Nyquist plot for a Randles cell (Figure 7) is always a semicircle. The solution resistance can be found by reading the real axis value at the high frequency intercept. This is the intercept near the origin of the plot. The real axis value at the other (low frequency) intercept is the sum of the polarization resistance and the solution resistance. The diameter of the semicircle is therefore equal to the polarization resistance (in this case 250 W).

Mixed Kinetic and Diffusion Control.

First consider a cell where semi-infinite diffusion is the rate determining step, with a series solution resistance as the only other cell impedance. A Nyquist plot for this cell is shown in Figure 9. Rs was assumed to be 20 W. The Warburg coefficient calculated to be about 120 Wsec-1/2 at room temperature for a two electron transfer, diffusion of a single species with a bulk concentration of 100 mM and a typical diffusion coefficient of 1.6×10-5 cm2/sec. Notice that the Warburg Impedance appears as a straight line with a slope of 45Â°.

Adding a double layer capacitance and a charge transfer impedance, we get the equivalent circuit in Figure 11.

This circuit models a cell where polarization is due to a combination of kinetic and diffusion processes, Figure 10.

Information theory is often considered to have begun with work by Harry Nyquist (H. Nyquist, Certain factors affecting telegraph speed, Bell System Technical Journal, 3, 324-346, 1924). While new knowledge is built by individuals standing on the shoulders of those who performed earlier research, people such as Nyquist can be seen as being extraordinarily creative for putting together previous work to produce a new and unique model.

Writing in the Bell System Technical Journal, Nyquist suggested that two factors determine the “maximum speed of transmission of intelligence”. Each telephone cable is implicitly considered to have a limit imposed on it such that there is a finite, maximum speed for transmitting “intelligence”.

This limit was widely understood by practicing electrical engineers of the era to be related to such factors as power, noise, and the frequency of the intelligent signal. Accepting such a limit as a given, Nyquist was able to work backwards towards the study of what was transmitted. He began referring to what was transmitted as “information.”

The two fundamental factors governing the maximum speed of data transmission are the shape of a signal and the choice of code used to represent the intelligence. Responding to the earlier work of Squier and others, Nyquist argues that telegraph signals are most efficiently transmitted when the intelligence carrying waves are rectangular. Given a particular “code”, use of square waves allows for intelligence to be transmitted faster than with sine waves in many practical environments.

Once the proper wave form is selected, a different problem arises: how should “intelligence” be represented? Telegraphers had long used Morse code and its variants to transmit text messages across distances. Each character was represented by a set of short or long electronic signals, the familiar dots and dashes. The letter C, for example, is represented in modern Morse code by a dash dot dash dot sequence.

Experienced telegraphers listen to messages at speeds far exceeding the ability of humans to consciously translate each individual dash or dot into a “thought representation” of the symbol; instead, Morse code is heard as a rhythm, with the rhythm for letters and common words being learned through long periods of listening.

Working backwards from the maximum telegraph speed, Nyquist considered the characteristics of an “ideal” code. Morse code is adequate for many applications, but an “adequate code” is far from being the best or optimal code available. Suggesting that the speed of intelligence transmission is proportional to the logarithm of the number of symbols which need to be represented, Nyquist was able to measure the amount of intelligence that can be transmitted using an ideal code. This is one step away from stating that there is a given amount of intelligence in a representation.

After his retirement, Nyquist was employed as a part time consultant engineer on communication matters by the Department of Defense, Stavid Engineering Inc., and the W.L. Maxson Corporation.

Before his death in 1976 Nyquist received many honors for his outstanding work in communications. He was the fourth person to receive the National Academy of Engineer’s Founder’s Medal, “in recognition of his many fundamental contributions to engineering.” In 1960, he received and the IRE Medal of Honor “for fundamental contributions to a quantitative understanding of thermal noise, data transmission and negative feedback.” Nyquist was also awarded the Stuart Ballantine Medal of the Franklin Institute in 1960, and the Mervin J. Kelly award in 1961. He passed away on 4 April 1976.

This work consists of a matrix of 12 x 16 (192) white LEDs, which display a portrait of the well-known engineer who was involved in analog-to-digital conversion theory. The portrait cycles through a sequence of random noise.