The Art of the Helicopter John Watkinson - Karatunov.net

Technical background 45
the cosine components will be in anti-phase and will cancel. Thus all real frequencies
actually contain equal amounts of positive and negative frequencies. These cannot be
distinguished unless a modulation process takes place. Figure 2.25(b) shows that when
two signals of frequency ±ω1 and ±ω2, are multiplied together, the result is that the
rotations of each must be added. The result is four frequencies, ±(ω1 +ω2) and ±(ω1 −
ω2), one of which is the sum of the input frequencies and one of which is the difference
between them. These are called sidebands. Sidebands are found extensively in avionics,
where the deliberate use of the process is called heterodyning. In a communications
radio the carrier frequency is multiplied or ‘modulated’ by the audio speech signal,
called the baseband signal, and the result is a pair of sidebands above and below the
carrier.
The rotation of the helicopter rotor has a certain frequency. Any vibration due to
periodic motion affecting the movement of the blades within the rotor plane may have
a characteristic frequency with respect to the rotor. When referred to the hull of the
helicopter, the frequency of the vibration may have been heterodyned by the rotor
frequency and the frequencies experienced in the hull may then be the frequencies of
the sidebands. This will be considered in Chapters 3 and 4.
In a voice radio system, the carrier frequency is much higher than the frequencies in
the speech, whereas in other systems this may not be the case. In digital systems, continuous
signals are represented by periodic measurements, or samples, and sidebands are
found above and below the frequency of the sampling clock Fs. Figure 2.26(a) shows
the spectrum of a sampling clock which contains harmonics because it is a pulse train
not a sinusoid. Figure 2.26(b) shows that the sidebands of the sampling frequency can
be rejected using a low-pass filter which allows through only the original baseband
signal. The sampled representation of the signal is returned to a continuous waveform.
This is exactly what happens in a Compact Disc player.
As the baseband frequency rises, the lower sideband frequency must fall. However,
this can only continue until the base bandwidth is half the sampling rate. This is known
as the Nyquist frequency and it represents the highest baseband frequency at which
the original signal can be recovered by a low-pass filter.
Fig. 2.26 (a) Spectrum of sampling pulses. (b) Spectrum of samples. (c) Aliasing due to sideband overlap.