dtft and ula: mathematical similarities of dtft spectrum and ula beampattern
TRANSCRIPT
Discrete-time fourier Tranform (DTFT) and UniformLinear Array (ULA):
mathematical similarities between the DTFT spectrumand the ULA beampattern.
C. J. Nnonyelu
PhD StudentDepartment 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
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
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
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
Discrete-Time Fourier Transform
Discrete-Time Fourier Transform Example
x(t) =
{1 , |t| ≤ 1
0 , otherwisex[n] =
{1 , n = 0,±1,±2
0 , otherwise
Figure 1: x(t), a continuous-time signal. Figure 2: x[n], the discrete-time sampleof x(t).
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 5 / 16
Discrete-Time Fourier Transform
Discrete-Time Fourier Transform Example
Xf (ω) =
2∑n=−2
e−jωn,
=sin(
5ω2
)sin(ω2
) .1 There are 5 discrete samples.
2 The argument of the sine function on the numerator is 5 times theargument of the sine function of the denominator.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 6 / 16
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
Uniform Linear Array
Simplifying assumptions and implications
1 Far-field wave, incident wave is streamlined at the point ofmeasurement (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 identicalproperties in all directions.
5 Co-planar wave, incident wave is on the same plane with the arrayhence one angle of consideration.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 8 / 16
Uniform Linear Array
A uniform linear array
Figure 4: A ULA with M identical isotropic sensors, aligned along the horizontalx-axis with a uniform separation of ∆ between adjacent sensors.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 9 / 16
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 isextra time taken relative to 0th sensor before wave arrives at the mthsensor.
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)mS(ω),
= ej2πλ
∆ cos(φ)mS(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
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
)]TS(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
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 andwavelength,
B(φ) = wH v(φ),
assuming S(ω) = 1. w := H(ω, φ, λ) ∈ CM×1 is a vector of complexweights.
C. J. Nnonyelu (PhD Student, EIE, HK PolyU) 14 September, 2014. 12 / 16
ULA Beamforming
ULA beampattern
If w = [wM−12, ..., w−1, w0, w1, ..., wM−1
2]T ,
B(φ) =
M−12∑
m=−M−12
w∗m ej 2π∆
λcos(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
Analogy between DTFT and ULA beampattern
Analogy between DTFT and ULA beampattern
B(φ) =
M−12∑
m=−M−12
w∗m ej 2π∆
λcos(φ) m, Xf (ω) =
+∞∑n=−∞
x[n] e−jωn
Signal’s discrete-time domain amplitudes x[n] equivalent to thesensors’ weighting w∗m. Identical sensors imply discrete-time samplesof equal amplitudes.
DTFT is continuous in ω ∈ [−π, π] (normalized frequency) and ULAis 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
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 designcan be adopted to calculate the weighting vector that would give adesired 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
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