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

Received Signal noise Attenuation Time delay 2

Modulation and Demodulation: one receiver

Recall: FT of Modulated Signal Usually

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

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

See a bit further … Call Then where Finally:

Beamforming Rec

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

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

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

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

Examples 13

Example: gain pattern of the array 14

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