Download - Odeon

upwards. This is the effect of the units being more than one wavelength apart from each
other. If the phase is shifted by one period, another angle of radiation is found. Geometrically,
the angles that correspond to i periods phase shift can be calculated from
c t
i c /
i
Arctg
d
f
where f is the frequency and i = ± 1, ± 2, etc.
The theoretical angles of radiation corresponding to this example are shown as a function of
the frequency in Table 11-2 below. The direction of the main lobe relative to the horizontal
direction is 10° for Δt = 0,1 ms and 19° Δt = 0,2 ms. The results for i = -1 explain the
upward side lobes seen at 2 and 4 kHz in fig. 11-4 and 11-5.
Table 11-2. Calculated angles of radiation from the example array with d = 0,20 m. Positive
angles are downwards and negative angles are upwards. Calculated for two different values of
the delay per unit, Δt.
Δt 0,1 ms Δt 0,2 ms
i 0 -1 1 i 0 -1 1
f (Hz) θ θ -1 θ 1 f (Hz) θ θ -1 θ 1
125 10 -86 86 125 19 -86 86
250 10 -82 82 250 19 -81 82
500 10 -73 75 500 19 -72 75
1000 10 -57 62 1000 19 -54 64
2000 10 -35 46 2000 19 -27 50
4000 10 -14 31 4000 19 -5 38
8000 10 -2 21 8000 19 7 29
Delay for transducers in an array should not be confused with delays assigned to the overall
delay assigned to a point source or array source in order to benefit from the HAAS effect,
although the devices used in order to obtain the delays may be identical:
Delays assigned in order to benefit from the HAAS effect are mainly supposed to have
effect in the time domain (because loudspeakers are supposed to have significant
distance between each other) and multiple reflections will blur the effect in the
frequency domain anyway.
Delays assigned to the individual transducers in an array on the other hand are mainly
supposed to have an effect in the frequency domain (the assumptions being that
transducers are closely spaced and that signals are periodic). Delaying a transducer
means that a signal will be emitted later from that transducer than it would otherwise.
Example:
If transducer 1 is delayed by 0 ms and transducer 2 is delayed by 3 ms, then at t = 3 ms
transducer 2 emits the same signal value as transducer 1 emitted at t = 0 ; by having a delay
the transducer looks back in time.
11.3 Playing with level
If the level is not the same for all the units in the array, but if it increases gradually from
bottom to top, the beam of radiation becomes asymmetric. An example is shown in fig. 11-6
with a level increase of 2 dB per unit.
11-103