Array Fundamentals and Conventional Spatial Filtering: beamforming

1. Array Fundamentals and Conventional Spatial Filtering: beamforming Director: Yufeng Zhang Reporter: Cuiqin Zhao No:1200500973

2. Contents • 1 Array Fundamentals • 1.1 Spatial Signals • 1.2 Array Signal Model • 1.3 Spatial Sampling • 2 Conventional Spatial Filtering: beamforming - Beam response • 2.1 Spatial Matched Filter -Element spacing -Array aperture and resolution • 3 Conclusions

3. 1 Array Fundamentals • Much as a frequency-selective filter emphasizes signals at a certain frequency ,we can choose to focus on signals from a particular direction. It has the ability to spatially discriminate, and passes signals from the certain directions while rejected those from other directions • Sensor array: spatial sampling of a spatially propagating signal.

4. 1.1 Spatial Signals(1) • Spatial signals are signals that propagate through space. A propagating wave emanating from a source located at : (1) Wavelength: (2)

5. 1.1 Spatial Signals(2) • Cone angle ambiguity surface for a uniform linear array (ULA) • The difference in propagation distance between neighboring elements: (3)

6. 1.2 Array Signal Model(1) • Block diagram of propagating signal arriving at a sensor with receive • The discrete-time signal form a ULA • The discrete-time model is (4) (5)

7. 1.2 Array Signal Model (2) • The full-array discrete-time model • Where the array response vector • The delay to the mth element with respect to the first element in the array • So (6) (7)

8. 1.3 Spatial sampling (8) • To avoid spatial alisaing • Spatial sampling frequency: • Frequency of propagating signal: • Normalized spatial frequency: • Using : • Since normalized frequency are unambiguous for and the full range of possible unambiguous angles is ,the sensor spacing must be (9) (10) (11) (12)

9. 2 Conventional Spatial Filtering: beamforming • Definition: to combine the signals from all the sensors in a manner, that is, with a certain weighting, so as to examine signals arriving from a specific angle. • The output is (13)

10. Beam response • Response for a given weight vector c: • Beam-pattern: • Convert spatial frequency to angle as:

11. 2.1 Spatial Matched Filter • Array response vector: the beam-forming weight vector from direction spatial matched filter: steering vector beam-former. The output of the spatial matched filter:

12. Element spacing d(1) • The beampatterns of spatial matched filters with for ULA with element spacing of , , and

13. Element spacing(2)

14. Array aperture and beamforming resolution • The aperture is the distance between the first and last elements. • Beam-forming resolution: the angular extent between the nulls of the mainbeam or the half-power points of the mainbeam (-3dB)

16. Conclusions • A brief background in some array fundamentals, including spatially propagating signals and array signal model. • Introduce the concept of beam-forming, that is ,the spatial discrimination or filtering of signals collected with a sensor array ,we look at conventional, that is, non-adaptive, beam-forming and touch upon many of the common considerations for an array that affect its performance, for example, element spacing, resolution, and side-lobe levels.

17. thanks