Now it's time to look at the theory that we need to implement for a ULA that enables us to figure out where signals come from and to listen to them. We are considering narrowband signals (i.e., sinusoids) of the form
x
(
t
)
=
ⅇ
j2πf
t
x
(
t
)
ⅇ
j2πf
t
(1)where f is the frequency of the signal. If we have N sensors numbered from n=0,...,N-1, then the delayed versions of x(t) that arrive at each microphone n are
x
n
(
t
)
=
ⅇ
j2πf
(
t
-
nτ
)
x
n
(
t
)
ⅇ
j2πf
(
t
-
nτ
)
(2)Thus, the first sensor (n=0) has zero delay, while the signal arrives at the second sensor (n=1) one unit delay later than at the first, and so on for each sensor. Then, we sample the signal at each microphone in the process of ADC, and call
x
n
(
r
)
=
x
n
(
mT
)
x
n
(
r
)
x
n
(
m
T
)
, where m is the integers. This gives us the sampled sinusoids at each sensor:
x
n
(
r
)
=
ⅇ
j2πf
(
r
-
n
τ
)
x
n
(
r
)
ⅇ
j2πf
(
r
-
n
τ
)
(3)Now we need to do some Fourier transforms to look at the frequency content of our received signals. Even though each sensor receives the same frequency signal, recall that delays
x
(
t
-
nτ
)
x
(
t
-
n
τ
)
in time correspond to modulation by
ⅇ
-
j
n
τ
ⅇ
-
j
n
τ
in frequency, so the spectra of the received signals at each sensor are not identical. The first Discrete Fourier Transform (DFT) looks at the temporal frequency content at each sensor:
X
n
(
k
)
=
1
R
∑
r
=
0
R
-
1
ⅇ
j2πf
(
r
-
n
τ
)
ⅇ
-
j2
π
kr
R
X
n
(
k
)
1
R
∑
r
=
0
R
-
1
ⅇ
j2πf
(
r
-
n
τ
)
ⅇ
-
j2
π
kr
R
(4)
ⅇ
-
j2
π
f
n
τ
R
ⅇ
-
j2
π
f
n
τ
R
for
k
fn
and
zow
(5)
for k=fn, and zero otherwise. Here we have used the definition of the normalized DFT, but it isn't particularly important whether you use the normalized or unnormalized DFT because ultimately the transform factors 1/Sqrt(R) or 1/R just scale the frequency coefficients by a small amount.
Now that we have N spectra from each of the array's sensors, we are interested to see how a certain frequency of interest fo is distributed spatially. In other words, this spatial Fourier transform will tell us how strong the frequency fo for different angles with respect to the array's broadside. We perform this DFT by taking the frequency component from each received signal that corresponds to fo and concatenating them into an array. We then zero pad that array to a length that is a power of two in order to make the Fast Fourier Transform (FFT) computationally efficient. (Every DFT that we do in this project is implemented as an FFT. We use the DFT in developing the theory because it applies always, whereas the FFT is only for computers.)
When we build the array of frequency components from each of the received signals, we have an N length array before we zero pad it. Let's think about the resolution of the array, which refers to its ability to discriminate between sounds coming from different angles. The greater the number of sensors in the array, the finer the array's resolution. Therefore, what happens when we zero pad the array of frequency components? We are essentially adding components from additional 'virtual sensors' that have zero magnitude. The result is that we have improved resolution! What effect does this have? Read on a bit!.
Once we have assembled our zero padded array of components fo, we can perform the spatial DFT:
Ω
(
k
)
=
1
NR
∑
n
=
0
N
-
1
ⅇ
-
j2
π
(
k
N
+
f
0
τ
)
Ω
(
k
)
1
NR
∑
n
=
0
N
-
1
ⅇ
-
j2
π
(
k
N
+
f
0
τ
)
(6)
where N for this equation is the length of the zero padded array and R remains from the temporal DFT. The result of the summation is a spectrum that is a digital sinc function centered at
f
0
τ
f
0
τ
. The value of the sinc represents the magnitude of the frequency fo at an angle theta. Because of the shape of the lobes of the sinc, which look like beams at the various angles, the process of using the array to look for signals in different directions is called
beamforming. This technique is used frequently in array processing and it is what enabled us to detect the directions from which certain frequency sounds come from and to listen in different directions.
Recalling that we zero padded our array of coefficients corresponding to
f
0
f
0
, what has that done for us in terms of the spatial spectrum? Well, we have improved our resolution, which means that the spectrum is smoother and more well-defined. This is because we are able to see the frequency differences for smaller angles. If we increase the actual number of sensors in the array, we will also improve our resolution and we will improve the beamforming by increasing the magnitude of the main lobe in the sinc spectrum and decreasing the magnitudes of the side lobes.
