1. Far Field (planar waves)
Propagation
A modulated signal can be transmitted through a medium
Transmitting Antenna
Near Field
Far Field (planar waves)
Receiving Antenna

2. Received Signal
noise
Attenuation
Time delay 2

3. Modulation and Demodulation: one receiver

4. Recall: FT of Modulated Signal
Usually

5. Multiple Receivers and a Planar Wave
We have a number of receivers
Rec
If the wave comes from direction we have:
with

6. See the FT of each received signal
Let with B the Bandwidth of the transmitted signal:
This implies:

7. See a bit further …
Call
Then
where
Finally:

8. Beamforming
Rec

9. If the waveform comes from a direction then
Beamforming
If the waveform comes from a direction then 9

10. Condition for no ambiguity
Since is periodic with period , in order to guarantee one to one correspondence between angle and gain we need to satisfy 10

11. Overall demodulated signal
Recall
With all phase shifts due to carrier propagation and synchronization errors.
Then, for a wavefront coming from direction 11

12. To maximize the return from a direction then choose the gains as
Choice of the Gains
To maximize the return from a direction then choose the gains as
where, again, 12

13. Examples 13

14. Example: gain pattern of the array14

15. You receive a signal from a direction There are receivers;
Example
You receive a signal from a direction
There are receivers;
Carrier Frequency
Then:
Wavelength:
Distance between receivers:
Spatial Frequency:
Gains: 15