# End-fire vs. Cardioid vs. Arc delay

A look at the pros and cons of each system

Now that we have a firm grasp of each of these building blocks for directional arrays, let’s look at the issues they have.

Cardioid

If we imagine a cardioid setup configuration as in Fig. 1, with the rear source reverse polarity and delay equal to the center-to-center propagation time for D.

Fig. 1: Cardioid Configuration

Let’s place observers both in front and behind and see what’s happening at those locations. First we’ll look at the front position. First arrival to this observer will be from the front source as it’s closest and has no delay. We’ll represent it as a sine wave, but any periodic signal will do. (See Fig. 2)

Fig. 2: First Source Arrival

At the forward observer point the wave front of the second source now arrives with reverse polarity. (See Fig. 3)

Fig. 3: Second Source Reverse Polarity

This second wave front arrives late by ¼ wavelength, which is equal to the separation distance D. (See fig. 4)

Fig. 4: Shift Due to Source Separation.

And finally when we add the electronic delay T, it shifts another ¼ wavelength back. (See Fig. 5)

Fig. 5: Wave Front Shift Due to T

So it’s now been shifted back a ½ wave length (D+T) and inverted, which is going to give us summation of +6 dB except for the first half cycle. (See Fig. 6)

Fig. 6: Summation

If we separate the sources by 1.0m (D), and add a delay of 2.902 msec (T), then the total delay would equal  5.814 msec (D+T) This means that for the first 5.184 msec the signal is at 0db, effectively only the first source is contributing here, and then the second source summation after this delay (D+T) causes the signal to rise to +6db.

Does this affect the transient envelope of the signal? I plan on trying this setup and measuring the impulse response to see what is happening to the transient envelope. (I suspect that at 80Hz and below it won’t make an audible difference.)

Let us now have a quick look at the rear observation point here in Fig. 7. We see first arrival from the rear source, the wave is out of polarity and travelling in the direction of the arrow.

Fig. 7: Rear Source Wave Front (reverse polarity)

Now the second wave front arrives from the front source but it’s delayed by distance D and its in polarity. (See Fig. 8)

Fig. 8: Front Source Delayed by Separation (D)

The first wave (rear source) is now delayed by time (T) (see Fig. 9), which produces cancellation (see Fig. 10).

Fig. 9: Rear Source Delayed by Time Delay (T)

Fig. 10: Cancellation

Cardioid arrays have perfect cancellation behind (within reason) but the possibility of transient smear in front. They are also very sensitive to boundary interference. See Fig. 11a, 11b, and 11c. Here a reflective surface is within two meters of the array and you can see how the directivity has been altered.

Fig. 11: Boundary Interference (a) 31Hz (b) 63Hz (c) 125Hz

End-Fire

The end-fire array works exactly in reverse to the cardioid array. In the forward direction the delay on the first source (T) matches the source separation (D) so the arrival time of the two sources are in time (phase) so we get summation (+6dB) but in the rearwards direction the first half wavelength is not cancelled, hence it’s leakage to the rear but it still achieves good rejection. So better transient-response at the expense of rear leakage.

Again these arrays are sensitive to boundaries (See Fig .12a, 12b, and 12c). Again, there is a reflective surface within 2m of the array and the altered directivity.

Fig. 12: Boundary Interference (a) 31Hz (b) 63Hz (c) 125Hz

Arc-Delay Processing Extreme Near-Field Issue
If an observer is within or less than half the effective radius of an arc-delay array he enters a region of cancellation.

If we take path length as our first arrival and then take the rest of the path length differences, b-a, c-a, d-a, and e-a, we can calculate these path length deltas in terms of time (see diagram 1 and table 1).

Table 1: Path Length Deltas in Msec

We can now plot the magnitude and phase of these delay interactions see figures 13-16

We can then perform an acoustic summation of these time offsets and plot the response at this point (see Fig. 17).

Fig. 17: Summed Response on 1/3 Octave Centers

As we can see a general broadband dip centered on 100 Hz, this has both good and bad attributes. Those people in the front row who are looking for “slam” or impact aren’t going to feel it; it only becomes apparent beyond the half- radius point from the array. The good part is, you don’t get extreme levels close to the system and the same happens behind so the “front-vocal line” is spared from excessive levels.

Conclusion:
Knowing what the pros and cons are of these arrays, allows us to pick the right tool for the job and also how to combine these types to our own devious ends. J

Next more complex arrays.