A short presentation made to highlight the similarities between the discrete-time fourier transform spectrum and uniform linear array beam pattern

1. Discrete-time fourier Tranform (DTFT) and Uniform
Linear Array (ULA):
mathematical similarities between the DTFT spectrum
and the ULA beampattern.
C. J. Nnonyelu
PhD Student
Department of Electronics and Information Engineering
Hong Kong Polytechnic University
14 September, 2014.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 1 / 16

2. Table of Contents
1 Purpose of presentation
2 Discrete-Time Fourier Transform
3 Uniform Linear Array
4 ULA Beamforming
5 Analogy between DTFT and ULA beampattern
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 2 / 16

3. Purpose of presentation
Purpose of Presentation
An introductory presentation to highlight:
1
Mathematical relationship between the DTFT
spectrum and the beampattern of the ULA.
2
How the similarities can benefit ULA design.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 3 / 16

4. Discrete-Time Fourier Transform
Discrete-Time Fourier Transform
Xf
(ω) =
+∞
n=−∞
x[n] e−jωn
,
where
x[n] is the discrete-time signal sample,
ω is the normalized angular frequency (normalized by the sample-rate)
with unit radian/sample,
n ∈ Z, set of integers.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 4 / 16

5. Discrete-Time Fourier Transform
Discrete-Time Fourier Transform Example
x(t) =
1 , |t| ≤ 1
0 , otherwise
x[n] =
1 , n = 0, ±1, ±2
0 , otherwise
Figure 1: x(t), a continuous-time signal. Figure 2: x[n], the discrete-time sample
of x(t).
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 5 / 16

6. Discrete-Time Fourier Transform
Discrete-Time Fourier Transform Example
Xf
(ω) =
2
n=−2
e−jωn
,
=
sin 5ω
2
sin ω
2
.
1 There are 5 discrete samples.
2 The argument of the sine function on the numerator is 5 times the
argument of the sine function of the denominator.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 6 / 16

7. Discrete-Time Fourier Transform
Discrete-Time Fourier Transform of x[n].
Figure 3: Xf
(ω) for x[n] with 5 samples.
1 Pattern repeats after 2π.
2 There are 4 lobes within every 2π span on the ω-axis.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 7 / 16

8. Uniform Linear Array
Simplifying assumptions and implications
1 Far-field wave, incident wave is streamlined at the point of
measurement (at the ULA).
2 Omnidirectional sensors, the sensitive is not dependent on direction.
3 Narrow-band signal, time-delays are approximated by a phase shift.
4 Homogeneous medium of propagation, medium has identical
properties in all directions.
5 Co-planar wave, incident wave is on the same plane with the array
hence one angle of consideration.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 8 / 16

9. Uniform Linear Array
A uniform linear array
Figure 4: A ULA with M identical isotropic sensors, aligned along the horizontal
x-axis with a uniform separation of ∆ between adjacent sensors.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 9 / 16

10. Uniform Linear Array
Deriving the array manifold of the ULA
The sensor located at (0, 0) i.e m = 0 is adopted as reference sensor.
Measurement at m = 0 is s(t), and s(t + τm) at mth sensor. τm is
extra time taken relative to 0th sensor before wave arrives at the mth
sensor.
In frequency-domain,
Sm(ω) = S(ω)ejω τm
.
τm = ∆ cos(φ)
c m, c is velocity of propagation (narrow-band),
Sm(ω) = ejω ∆
c
cos(φ)m
S(ω),
= ej 2π
λ
∆ cos(φ)m
S(ω)
since c = fλ, and ω = 2πf. λ is the wavelength of the incident wave.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 10 / 16

11. Uniform Linear Array
The ULA array manifold
Collection of all sensors’ measurements
S(ω, φ, λ) = exp j
2π
λ
∆ cos(φ) −
M − 1
2
, ..., −1, 0, 1, ...,
M − 1
2
T
S(ω)
Assuming S(ω) = 1, the array manifold
v(ω, φ, λ) = exp j
2π
λ
∆ cos(φ) −
M − 1
2
, ..., −1, 0, 1, ...,
M − 1
2
T
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 11 / 16

12. ULA Beamforming
Array frequency response and beampattern
The beamformer’s output in frequency domain, i.e its frequency response
Y (ω, φ, λ) = H(ω, φ, λ) · S(ω, φ, λ),
H(ω, φ, λ) is the filter which the received signal is passed through.
Beampattern is the frequency response to a wave of specific frequency and
wavelength,
B(φ) = wH
v(φ),
assuming S(ω) = 1. w := H(ω, φ, λ) ∈ CM×1 is a vector of complex
weights.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 12 / 16

13. ULA Beamforming
ULA beampattern
If w = [wM−1
2
, ..., w−1, w0, w1, ..., wM−1
2
]T ,
B(φ) =
M−1
2
m=−M−1
2
w∗
m ej 2π ∆
λ
cos(φ) m
w represents the beamformer, e.g.
1 delay-and-sum beamformer.
2 spatial matched beamformer.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 13 / 16

14. Analogy between DTFT and ULA beampattern
Analogy between DTFT and ULA beampattern
B(φ) =
M−1
2
m=−M−1
2
w∗
m ej 2π ∆
λ
cos(φ) m
, Xf
(ω) =
+∞
n=−∞
x[n] e−jωn
Signal’s discrete-time domain amplitudes x[n] equivalent to the
sensors’ weighting w∗
m. Identical sensors imply discrete-time samples
of equal amplitudes.
DTFT is continuous in ω ∈ [−π, π] (normalized frequency) and ULA
is continuous in φ ∈ [−π, π] (angle of arrival - spatial frequency).
DTFT is 2π periodic while ULA is π periodic.
Summarily,
[w]m ≡ x∗
[n],
2π ∆
λ cos(φ) ≡ ω.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 14 / 16

15. Analogy between DTFT and ULA beampattern
Implications of similarities
1 If ∆
λ = 1
2, then 2π∆
λ cos(φ) ∈ [−π, π] which would be same for
ω ∈ [−π, π].
2 Under certain conditions, the window techniques used in filter design
can be adopted to calculate the weighting vector that would give a
desired beampattern with aim at achieving a desired
1 mainlobe beamwidth,
2 mainlobe-to-highest-sidelobe height ratio,
3 null positions,
4 mainlobe location.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 15 / 16

16. Analogy between DTFT and ULA beampattern
Consulted Text(s)
H. L. Van Trees, “Detection, Estimation, and Modulation Theory,
Part IV, Optimum Array Processing ,” New York: Wiley, 2004.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 16 / 16