Heterodyne

This article is about waveform manipulation. For the use of the term in poetry, see Heterodyne (poetry). For the webcomic characters, see Girl Genius.

Heterodyning is a radio signal processing technique invented in 1901 by Canadian inventor-engineer Reginald Fessenden where high frequency signals are converted to lower frequencies by combining two frequencies.^[1] Heterodyning is useful for frequency shifting information of interest into a useful frequency range following modulation or prior to demodulation. The two frequencies are combined in a vacuum tube, transistor, diode, or other non-linear signal processing device. Heterodyning creates two new frequencies, according to the properties of the sine function; one is the sum of the two frequencies mixed, the other is their difference. These new frequencies are called heterodynes. Typically only one of the new frequencies is desired—the higher one after modulation, and the lower one after demodulation. The other signal is filtered out of the output of the mixer.

History

The heterodyne technique was pioneered by Canadian inventor-engineer Reginald Fessenden in 1901, but was not pursued very far because the local oscillators being used at the time were unstable.^[2] The technique was invented as a means to make Morse code radiotelegraph (Continuous wave) signals used during the wireless telegraphy era audible.^[3] A "heterodyne" or "beat" receiver has a local oscillator (LO), that produces a radio signal adjusted to be close in frequency to the incoming signal being received. When the two signals are mixed the difference or a "beat" frequency exists in the audible range. This conjoint action of two radio-frequency oscillations produces a musical tone in a telephonic receiver or loud speaker. The Morse code "dots" and "dashes" are audible as beeping sounds. This technique is still used in radio telegraphy, the local oscillator now being called the beat frequency oscillator or BFO. Fessenden coined the word heterodyne from the Greek roots hetero- "different", and dyn- "power" (cf. dynamis).^[4]

An improvement on the heterodyne receiver is the superheterodyne receiver (superhet), invented by Edwin Howard Armstrong in 1918. It converts the incoming Radio Frequency (RF) signal to a fixed Intermediate Frequency (IF), using the heterodyne technique. The difference between the superhet and Fessenden's heterodyne is the use of a tunable RF filter on the front end, a mixer circuit, a stable local oscillator, and a fixed frequency high-gain band-pass amplifier.^[5] The original heterodyne technique tried to accomplish all of this in one stage thus producing an unstable amplifier.

Applications

Heterodyning is used very widely in communications engineering to generate new frequencies and move information from one frequency channel to another. Besides its use in the superheterodyne circuit which is found in almost all radio and television receivers, it is used in radio transmitters, modems, satellite communications and set-top boxes, radar, radio telescopes, telemetry systems, cell phones, cable television converter boxes and headends, microwave relays, metal detectors, atomic clocks, and military electronic countermeasures (jamming) systems.

Up and down converters

In digital communications signals can be transmitted in baseband or passband. Typically, a down-converter is used on the receiving end to transform the signal from the passband back to the baseband for further processing. The local oscillator frequency is $\sqrt{2} e^{j2 \Pi f_c t}$, with f_c being the carrier frequency. This results in a scaling of the received signal by $\sqrt{2}$ and a phase shifting by f_c to the left, so that the resulting signal is located in the baseband.

A radio frequency upconverter is a device that takes an input of radio frequency energy of a specific frequency range and outputs it on a higher frequency. Likewise, downconverters take an input frequency and reduce it to a lower output frequency. Both converters are commonly used in transverters and satellite communications. Upconverters achieve this frequency conversion via heterodyning, the same principle as modern receivers and transmitters to offset the frequency.

Analog videotape recording

Many analog videotape systems rely on a downconverted color subcarrier in order to record color information in their limited bandwidth. These systems are referred to as "heterodyne systems" or "color-under systems". For instance, for NTSC video systems, the VHS (and S-VHS) recording system converts the color subcarrier from the NTSC standard 3.58 MHz to ~629 kHz.^[6] PAL VHS color subcarrier is similarly downconverted (but from 4.43 MHz). The now-obsolete 3/4" U-matic systems use a heterodyned ~688 kHz subcarrier for NTSC recordings (as does Sony's Betamax, which is at its basis a 1/2" consumer version of U-matic), while PAL U-matic decks came in two mutually incompatible varieties, with different subcarrier frequencies, known as Hi-Band and Low-Band. Other videotape formats with heterodyne color systems include Video-8 and Hi8.^[7]

The heterodyne system in these cases is used to convert quadrature phase-encoded and amplitude modulated sine waves from the broadcast frequencies to frequencies recordable in less than 1 MHz bandwidth. On playback, the recorded color information is heterodyned back to the standard subcarrier frequencies for display on televisions and for interchange with other standard video equipment.

Some U-matic (3/4") decks feature 7-pin mini-DIN connectors to allow dubbing of tapes without a heterodyne up-conversion and down-conversion, as do some industrial VHS, S-VHS, and Hi8 recorders.

Music synthesis

The theremin, an electronic musical instrument, uses the heterodyne principle to produce a variable audio frequency in response to the movement of the musician's hands in the vicinity of some antennas. The output of a fixed radio frequency oscillator is mixed with that of an oscillator whose frequency is affected by the variable capacitance between the antenna and the thereminist as that person moves her or his hand near the pitch control antenna. The difference between the two oscillator frequencies produces a tone in the audio range.

The Ring modulator is a type of heterodyne incorporated into some synthesizers or used as a stand-alone audio effect.

Optical heterodyning

Optical heterodyne detection (an area of active research) is an extension of the heterodyning technique to higher (visible) frequencies. This technique could greatly improve optical modulators, increasing the density of information carried by optical fibers. It is also being applied in the creation of more accurate atomic clocks based on directly measuring the frequency of a laser beam.^[8]

Since optical frequencies are far beyond the manipulation-capacity of any feasible electronic circuit, all photon detectors are inherently energy detectors not oscillating electric field detectors. However, since energy detection is inherently "square-law" detection, it intrinsically mixes any optical frequencies present on the detector. Thus, sensitive detection of specific optical frequencies necessitates optical heterodyne detection, in which two different (close-by) wavelengths of light illuminate the detector so that the oscillating electrical output corresponds to the difference between their frequencies. This allows extremely narrow band detection (much narrower than any possible color filter can achieve) as well as precision measurements of phase and frequency of a light signal relative to a reference light source, as in Laser Doppler Vibrometry.

This phase sensitive detection has been applied for Doppler measurements of wind speed, and imaging through dense media. The high sensitivity against background light is especially useful for LIDAR.

In optical Kerr effect (OKE) spectroscopy, optical heterodyning of the OKE signal and a small part of the probe signal produces a mixed signal consisting of probe, heterodyne OKE-probe and homodyne OKE signal. The probe and homodyne OKE signals can be filtered out, leaving the heterodyne signal for detection.

Mathematical principle

Heterodyning is based on the trigonometric identity:

$\sin \theta \sin \varphi = \frac{1}{2}\cos(\theta - \varphi) - \frac{1}{2}\cos(\theta + \varphi)$

The product on the left hand side represents the multiplication ("mixing") of a sine wave with another sine wave. The right hand side shows that the resulting signal is the difference of two sinusoidal terms, one at the sum of the two original frequencies, and one at the difference, which can be considered to be separate signals.

Using this trigonometric identity, the result of multiplying two sine wave signals, $\sin (2 \pi f_1 t)\,$ and $\sin (2 \pi f_2 t)\,$ can be calculated:

$\sin (2 \pi f_1 t)\sin (2 \pi f_2 t) = \frac{1}{2}\cos [2 \pi (f_1 - f_2) t] - \frac{1}{2}\cos [2 \pi (f_1 + f_2) t] \,$

The result is the sum of two sinusoidal signals, one at the sum f₁ + f₂ and one at the difference f₁ - f₂ of the original frequencies

The two signals are multiplied in the mixer. In order to multiply the signals, the mixer must be a nonlinear component, that is, its output current or voltage must be a nonlinear function of its input. Most circuit elements in communications circuits are designed to be linear. This means they obey the superposition principle; if F(v) is the output of a linear element with an input of v:

$F(v_1 + v_2) = F(v_1) + F(v_2) \,$

So if two sine wave signals are applied to a linear device, the output is simply the sum of the outputs when the two signals are applied separately, with no product terms. So the function F must be nonlinear. Examples of nonlinear components that are used as mixers are vacuum tubes and transistors biased near cutoff (class C), and diodes. For lower frequencies, IC analog multipliers can be used which multiply signals precisely. Ferromagnetic core inductors driven into saturation can also be used. In nonlinear optics, crystals that have nonlinear characteristics are used to mix laser light beams to create heterodynes at optical frequencies.

To demonstrate mathematically how a nonlinear component can multiply signals and generate heterodyne frequencies, the nonlinear function F can be expanded in a power series (MacLaurin series):

$F(v) = \alpha_1 v + \alpha_2 v^2 + \alpha_3 v^3 + \ldots \,$

To simplify the math, the higher order terms above α₂ will be indicated by an ellipsis (". . .") and only the first terms will be shown. Applying the two sine waves at frequencies ω₁ = 2πf₁ and ω₂ = 2πf₂ to this device:

$v_{out} = F(A_1 \sin \omega_1 t + A_2 \sin \omega_2 t)\,$

$v_{out} = \alpha_1 (A_1 \sin \omega_1 t + A_2 \sin \omega_2 t) + \alpha_2(A_1 \sin \omega_1 t + A_2 \sin \omega_2 t)^2 + \ldots \,$

$v_{out} = \alpha_1 (A_1 \sin \omega_1 t + A_2 \sin \omega_2 t) + \alpha_2(A_1^2 \sin^2 \omega_1 t + 2 A_1 A_2 \sin \omega_1 t \sin \omega_2 t + A_2^2 \sin^2 \omega_2 t) + \ldots \,$

It can be seen that the second term above contains a product of the two sine waves. Simplifying with trigonometric identities:

$v_{out} = \alpha_1 (A_1 \sin \omega_1 t + A_2 \sin \omega_2 t) + \alpha_2(\frac{A_1^2}{2} [1 - \cos 2 \omega_1 t] + A_1 A_2 [\cos (\omega_1 t - \omega_2 t) - \cos (\omega_1 t + \omega_2 t) ] + \frac{A_2^2}{2} [1 - \cos 2 \omega_2 t] ) + \ldots \,$

$v_{out} = \alpha_2 A_1 A_2 \cos (\omega_1 - \omega_2 )t - \alpha_2 A_1 A_2 \cos (\omega_1 + \omega_2 ) t + \ldots \,$

So the output contains sinusoidal terms with frequencies at the sum ω₁ + ω₂ and difference ω₁ - ω₂ of the two original frequencies. It also contains terms at the original frequencies and at multiples of the original frequencies 2ω₁, 2ω₂, 3ω₁, 3ω₂, etc.; the latter are called harmonics. These unwanted frequencies, along with the unwanted heterodyne frequency, must be filtered out of the mixer output to leave the desired heterodyne.

See also

• Optical heterodyne detection
• Beat (acoustics)
• Edwin Howard Armstrong
• Electroencephalography
• Heterodyne detection
• Homodyne
• Superheterodyne receiver
• Transverter
• Intermodulation - a problem with strong higher-order terms produced in some non-linear mixers

Notes

1. ^ Christopher E. Cooper, Physics, page 25
2. ^ Nahin, Paul. The Science of Radio. Page 91. Figure 7.10. Chapter 7. ISBN 0-387-95150-4.
3. ^ Ashley, Charles Grinnell; Charles Brian Heyward (1912). Wireless Telegraphy and Wireless Telephony. Chicago: American School of Correspondence. pp. 103/15–104/16. http://books.google.com/books?id=pK-EAAAAIAAJ&pg=PA103. 
4. ^ Tapan K. Sarkar, History of wireless, page 372
5. ^ Nahin, Paul. The Science of Radio. Page 285. Chapter 21. ISBN 0-387-95150-4.
6. ^ Videotape formats using ¹⁄₂-inch-wide (13 mm) tape ; Retrieved 2007-01-01
7. ^ Poynton, Charles. Digital Video and HDTV: Algorithms and Interfaces San Francisco: Morgan Kaufmann Publishers, 2003 PP 582, 583 ISBN 1-55860-792-7
8. ^ http://tsapps.nist.gov/ts_sbir/abstracts/08abst1.htm see NIST subtopic 9.07.9-4.R for a description of research on one system to do this.

References

• Glinsky, Albert. Theremin: Ether Music and Espionage. Urbana: University of Illinois Press, 2000. ISBN 0-252-02582-2.
• Nahin, Paul J. The Science of Radio. New York: Springer-Verlag, AIP Press, 2001. ISBN 0-387-95150-4.