This page is ©2004-2022 F. Dörenberg, unless stated otherwise. All rights reserved worldwide. No part of this publication may be used without permission from the author.

Latest page update: 26 August 2022

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Bandwidth ... often a controversial topic. When operating a transmitter, knowing (or at least being aware of) your bandwidth is just as important as knowing your operating frequency. Part of the controversy stems from the definition of "bandwidth". Often, the term is qualified with "necessary", "occupied", "actual", "effective", "specified", etc., or is not qualified at all. Either way, a proper definition should refer to power levels (ref. 1):

The fractional power containment bandwidth is the definition of bandwidth that has been most used to define channel bandwidth for digital modulation.... It is the most appropriate measure of necessary bandwidth because it is a measure of the integrated power spectrum density, and can be related to system performance.

The CCIR/ITU and the ARRL (ref. 2, 8D) use the definition of "99% power bandwidth":

Bandwidth -- The width of a frequency band outside of which the mean power is attenuated at least 26 dB below the mean power of the total emission, including allowances for transmitter drift or Doppler shift. Bandwidth describes the range of frequencies that a radio transmission occupies. [26 dB = 0.5% beyond each bandwidth limit]

The IARU gives an example of "transmitted signal bandwidth" (ref. 3):

The -6 dB bandwidth of a properly shaped CW signal is approximately 4 times the sending speed in WPM (Words Per Minute). Example: CW at 25 WPM takes 100 Hz (at -6 dB).


Let us take a tone with frequency f0 and key that tone on/off with a square wave that has a period 2T. The frequency spectrum of the resulting tone-pulse stream consists of a spectral line at f0, and an infinite Fourier-series of sideband lines at all odd multiples N of 1/(2T) on both sides of f0. The height of the sideband lines decreases with K, via a (1/Kπ)2 relationship. See the left-hand image in Fig. 1. Keying a tone on/off with a continuous square wave signal is not particularly useful for conveying information. If the tone is keyed with a (quasi) random sequence of on/off pulses with duration T, then the sideband spectrum lines are "smeared" into half sine-wave envelopes. See the right-hand image in Fig. 1. This is basically the spectrum of Hellschreiber tone-pulses.

Bernhard transmitters

Fig. 1: spectrum of a tone that is keyed on & off

(left: keying with a square wave with 50% duty-cycle; right: keying with random sequence of such on/off pulses)

The shape and the extent of the spectrum of an on/off-keyed (OOK) signal depends two primary factors:

  • the duration T of the fundamental "on" and "off" pulses.
  • the shape of the envelope of the pulses. The extreme case is a perfectly rectangular pulse. It has an infinitely steep discontinuity at its beginning and end, and its spectrum includes an infinite number of harmonics. Obviously, in real life, there are no switching and other engineering components that are infinitely fast. So, there is always some inherent bandwidth limiting throughout a system (amplifiers, circuitry for grid, anode, or cathode keying of vacuum tubes, etc.). However, this inherent limiting typically results in transmission of a signal that has a bandwidth that is well beyond what is necessary and allowed or desired.

If the carrier of a CW telegraphy transmitter is keyed on/off with a stream of rectangular pulses (such as the envelope of tone pulses), then the RF spectrum of the transmitted signal shows a large very narrow peak at the carrier frequency. Symmetrically about this carrier frequency, there is a series of side-lobes. See Fig. 2 below.


Fig. 2: spectrum of the RF output signal of a CW-telegraphy transmitter, keyed with a stream of rectangular pulses

If the carrier of an AM transmitter is modulated with a stream of rectangular tone pulses, then the RF spectrum of the transmitted signal shows a large very narrow peak at the carrier frequency, and two identical sidebands, one on each side of the carrier frequency, at a distance equal to the tone frequency. See Fig. 3 below. The sidebands are mirror-imaged, but since they are symmetrical, they are the same.


Fig. 3: spectrum of the RF output signal of an AM transmitter, modulated with rectangular tone pulses

In a decent SSB (Single Side Band - Suppressed Carrier) transmitter, the carrier and the unwanted sideband are suppressed by at least 40 dB. The resulting RF spectrum for modulation with a stream of rectangular tone pulses, is effectively the same as that of a CW transmitter that is keyed with the same pulses. However, compared to the CW spectrum, the SSB sideband spectrum is shifted to the left (Lower Sideband, LSB) or right (Upper Sideband, USB) by a distance that is equal to the tone frequency.


Fig. 4: spectrum of the RF output signal of an SSB transmitter (in USB mode), modulated with rectangular tone pulses


Clearly, transmitting a signal that occupies more bandwidth than needed for the intended communication, is highly undesirable, as it causes adjacent channel interference, out-of-band transmission, etc. Therefore, this is usually not allowed. So, we have to reduce the occupied bandwidth that results from on/off-keying.

Siemens-Halske, the Cable & Wireless company of the UK, and the Reichspostzentralamt (central office of the German national postal authority) conducted extensive Hellschreiber experiments and measurements around 1935 (ref. 4). This resulted in the recommendation for filtering the modulation (i.e., at the transmitter) with a low-pass filter that has a corner frequency of 1.2 times the pixel rate. This effectively suppresses the third and higher harmonics of the pixel rate, and significantly reduces the second harmonic. This filtering still allows the text that is printed at the receiving Hellschreiber, to be perfectly legible. This can actually be verified by using a very narrow IF or AF filter at the receiver. Of course, filtering at the receiver does not affect the bandwidth of the transmitted signal, and may worsen print quality.

The font used by the Feld-Hell machines has black and white column segments ( = back-to-back pixels)  that have a duration of N x 1000 msec / (2.5 characters/sec x 7 columns x 14 lines) = N x 4.08 msec, where N is an integer value of at least 2. I.e., 8.16 msec, 12.24 msec, 16.32 msec, etc. Hence, the shortest pixel cycle ( = 1 black + 1 white pixel) is 2 x 8.16 = 16.32 msec. This equivalent to a pixel rate ("Punktfrequenz"; ref. 5) of 1000 / 16.32 = 61.25 Hz. As the Hell-pixels are binary, the equivalent "telegraphy speed" is 1000 / 8.16 = 122.5 baud. "Presse Hell" uses the same font, but at twice the speed, hence, half the fundamental pulse duration.

The above companies furthermore recommended against widening the transmitter's output filter beyond 1.6 times the pixel rate, i.e., 1.6 x 61.25 ≈ 100 Hz at 2.5 characters/sec (Feld-Hell), or 200 Hz at 5 characters/sec (Presse-Hell). Ref. 6, 16. Actually implemented low-pass filters of the era also blocked frequencies above 200 Hz (ref. 7). As usual, the referenced literature does not indicate steepness of the applied filters. Note that at that time, the corner frequency of a filter was defined as being 4.34 dB ( = 0.5 neper) below the passband, whereas in modern times, a factor of 3 dB is used. The above factor 1.6 is also supported by Shannon's theorem (and the Hartley-Shannon law) regarding the maximum data rate for "near error-free" communication across a bandwidth limited channel. It is used in the table below (though normally applied to the Baud-rate).

These experiments and analyses led to the official acceptance of Hellschreiber-telegraphy at the International Radiocommunications Conference held in Cairo in 1938, for use on frequencies reserved for A1-modulated telegraphy (ref. 8A-8C).


Fig. 5: Shannon bandwidth for Hellschreibers with two and four times the speed of Feld-Hell

(source: p. 65 in ref. 9A; same results shown in ref. 9B, 10)

The next table captures the definition of the FCC (ref. 11) and the ITU for "necessary bandwidth". The formulas date back at least to the Joint Telegraph and Radiotelegraph Conference in Madrid (1932), as confirmed at the 1938 International Radiocommunications Conference. For Feld-Hell, the "necessary bandwidth", in combination with a non-fading transmission path, is 3 x 122.5 ≈ 367 Hz. Ref. 12 . The factor 3 is used in order to be able to preserve a reasonably shaped square wave (ref. 13), and this requires at least the third harmonic of the pixel rate (ref. 14).


Fig. 6: Necessary bandwidth of amplitude modulated signal with quantized or digital information

(note: the formula for tone modulated carriers applies to AM (double sideband) modulation)

In the above table, the formula for tone modulated carrier applies to AM modulation: double sideband, unsuppressed carrier. Hence, the formula includes 2 x fmodulation, to account for there being two sidebands. However, the spectrum of an on-off keyed single-tone modulated carrier with SSB modulation (single-sideband with suppressed carrier) is basically indistinguishable from the spectrum of an on-off keyed carrier (CW). The real Feld-Hell machines can operate in two on-off keyed modes: 1) direct keying of a CW transceiver, and 2) keying of an internally generated constant 900 Hz signal, in combination with an voice/telephony transceiver (AM, SSB).

Note that "necessary bandwidth" is absolutely not the same as "occupied bandwidth"! The latter depends on the shape of the pulse envelope, resulting from the (CW) transmitter's keying characteristics or applied by DSP/software processing ("raised-cosine" comes close to the ideal shape). Improper pulse shaping may cause the actual bandwidth to be several 10s of kilohertz under good propagation conditions! Obviously overdriving/over-modulating the transmitter will also result in (much) higher actual signal bandwidth (many kHz!). Unfortunately this is not at all uncommon for operators with PC-Hellschreibers (or digi-modes in general) that use a PC-soundcard...

Note that the Nyquist minimum bandwidth for Feld-Hell is 2 x 122.5 = 245 Hz. This is the theoretical minimum bandwidth that can be used to represent a signal and still allow loss-less reconstruction after transmission over an ideal, noiseless transmission channel. Shaping a rectangular pulse with (root) raised-cosine (or Gaussian) filters allow this minimum bandwidth to be closely approximated. The frequency response of raised-cosine filters is flat in the band-pass, follows a gradual s-curve at the leading and trailing edges of the band-pass, and is zero outside the band-pass. Discrete implementations of such filters are common place in portable phones and modems. But today's digi-mode software is also fully capable of providing such approximations. Software-implemented Feld-Hell (e.g., IZ8BLY, DM780, FLdgi and MULTIPSK) typically applies raised-cosine pulse shaping - except, of course, for FM-modes. Ref. 17.


Fig. 7: tone pulses with raised-cosine pulse shaping

Based on all of the above formulas, Feld-Hell has a necessary bandwidth of 245-367 Hz. This is well below 500 Hz, hence qualifies as a "narrow bandwidth" mode. This may be compared to the bandwidth of an other popular narrow digital amateur radio mode: PSK31. It has a specified bandwidth of 62.5 Hz at -30 dB (in practice about 80 Hz). PC-software Hellschreiber using a soundcard and an SSB transceiver, should produce a transmitted signal with a bandwidth per the various formulas stated above.

Can we "do" Hellschreiber with less bandwidth? Well... yes. That is the beauty and strength of this system! The original Hell font, combined with the pattern-recognition capability of the human brain, allows us to read Hell text well into the noise level, and with as much as 60% distortion due to pixel shortening or lengthening ("smearing"), see ref. 4. We do not need loss-less reconstruction of the transmitted signals. An otherwise clean signal, may be readable down to, or below, a bandwidth equal to the reciprocal of the baud rate, i.e., ≤ 122.5 Hz.


Fig. 8: audio spectrum of Feld-Hell, RTTY, and PSK-31 (shown from 600-1300 Hz, centered on 1000 Hz)

(bandwidth-limiting raised-cosine pulse-shaping applied!!!)


Feld-Hell is typically reported as having a duty cycle of 22%. This is based on the word "HELL" having a duty-cycle of 22.5%. Note that:

  • it is only the duty-cycle of this specific word.
  • it is the average duty cycle of that word.
  • it is only the duty-cycle of the original Hell-font. Some Hellschreiber software has the option to use a "DX" version of the Hell-font (or whatever font is selected): it is double-wide, and all columns are sent twice. This does not change the average duty cycle. PC-fonts have a different duty-cycle - and should not be used anyway.
  • Hell-modes other than Feld-Hell (i.e., Hell-80, FM-Hell, PSK-Hell) have a duty cycle of 100% - independent of the font that is used. FSK-Hell has an average duty-cycle of about 80%.

But yes: the Feld-Hell font does have a low average duty-cycle, which makes it easy on the transmitter. My binary file of the Feld-Hell font shows that the character "8" has the highest duty-cycle: 39% (it consists of 38 out of the 7x14=98 possible pixel-bits). The "period" (a.k.a. "full stop") has the lowest duty-cycle: 6% (6 pixels total out of 98).


External links last checked: April 2016

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©2004-2016 F. Dörenberg, unless stated otherwise. All rights reserved worldwide. No part of this publication may be used without permission from the author.