dtft and ula: mathematical similarities of dtft spectrum and ula beampattern

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


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.


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.



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).



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.



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.


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.



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.



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.



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


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.


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.


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.



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.



Consulted Text(s)

H. L. Van Trees, “Detection, Estimation, and Modulation Theory,Part IV, Optimum Array Processing ,” New York: Wiley, 2004.
