OFDM (Orthogonal Frequency Division Multiplexing)

Channel multipath 𝐻 Tx Rx Channel Impulse Response Amplitude time 𝐹 𝑐 =𝑓 𝐹 𝑐 =−𝑓 Channel Frequency Response 𝐻 𝑓

Received data with channel multipath 𝑋 𝑓 𝐹 𝑐 =−𝑓 𝐹 𝑐 =𝑓 𝐹 𝑐 =𝑓 𝐹 𝑐 =−𝑓 𝐻 𝑓 𝐹 𝑐 =𝑓 𝐹 𝑐 =−𝑓 𝑌=𝐻∗𝑋 𝑌 𝑓 = 𝐻 𝑓 𝑋 𝑓

A few frequencies are completely faded Deep channel fading 𝑋 𝑓 𝐹 𝑐 =𝑓 𝐹 𝑐 =−𝑓 𝐻 𝑓 𝐹 𝑐 =−𝑓 𝐹 𝑐 =𝑓 A few frequencies are completely faded 𝐹 𝑐 =𝑓 𝐹 𝑐 =−𝑓 𝑌=𝐻∗𝑋 𝑌 𝑓 = 𝐻 𝑓 𝑋 𝑓 Whole message can be destroyed even if few frequencies of the message are lost

Revisit the transmitted spectrum F -F

Increase the symbol duration Symbol rate R s =2∗𝐵 (bandwidth) F -F Decreases the bandwidth of the signal -F F

Modified signal passed through the channel 𝑋 𝑓 𝐹 𝑐 =−𝑓 𝐹 𝑐 =𝑓 𝐻 𝑓 𝐹 𝑐 =−𝑓 𝐹 𝑐 =𝑓 𝐹 𝑐 =𝑓 𝐹 𝑐 =−𝑓 Data is unaffected since the fading frequencies do not overlap with data frequencies

Coping with fading Decrease bandwidth so that data frequencies do not overlap with fading frequencies This helped eliminate the effect of fading Disadvantage: This would waste a lot of available bandwidth Can we do better to achieve throughput proportional to the channel quality, without wasting any bandwidth

The message is chunked into groups of sub-messages Coping with fading 𝑥 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑥 The message is chunked into groups of sub-messages F -F F -F 𝑥 1 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑥 2 𝑥 3 𝑥 4

Coping with fading 𝑒 𝑖2𝜋 𝑓 1 𝑡 𝑒 𝑖2𝜋 𝑓 2 𝑡 𝑒 𝑖2𝜋 𝑓 3 𝑡 𝑒 𝑖2𝜋 𝑓 4 𝑡 x x 𝑥 𝑥 1 𝑥 2 𝑥 3 𝑥 4 F -F 𝑓 1 𝑓 2 𝑓 3 𝑓 4 𝑥 1 𝑥 2 𝑥 3 𝑥 4 x 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑒 𝑖2𝜋 𝑓 1 𝑡 x 𝑒 𝑖2𝜋 𝑓 2 𝑡 x 𝑒 𝑖2𝜋 𝑓 3 𝑡 x 𝑒 𝑖2𝜋 𝑓 4 𝑡

Coping with fading + + + 𝑒 𝑖2𝜋 𝑓 1 𝑡 𝑒 𝑖2𝜋 𝑓 2 𝑡 𝑒 𝑖2𝜋 𝑓 3 𝑡 x x x x F 𝑥 𝑥 𝑥 1 𝑥 2 𝑥 3 𝑥 4 x + 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑒 𝑖2𝜋 𝑓 1 𝑡 x 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑒 𝑖2𝜋 𝑓 2 𝑡 x 𝑓 4 𝑓 3 𝑓 2 𝑓 1 + 𝑒 𝑖2𝜋 𝑓 3 𝑡 + x

Sub-messages are loaded on different frequencies 𝑥 1 𝑥 2 𝑥 3 𝑥 4 F -F x 𝑥(𝑡) 𝑥 1 (𝑡) 𝑒 𝑖2𝜋 𝑓 1 𝑡 + x 𝑥 2 (𝑡) 𝑒 𝑖2𝜋 𝑓 2 𝑡 + 𝑥 1 𝑥 2 𝑥 3 𝑥 4 x 𝑥 3 (𝑡) 𝑓 4 𝑓 3 𝑓 2 𝑓 1 𝑒 𝑖2𝜋 𝑓 3 𝑡 + x Sub-messages are loaded on different frequencies 𝑥 4 (𝑡) 𝑒 𝑖2𝜋 𝑓 4 𝑡

𝑥 1 𝑥 2 𝑥 3 𝑥 4 F -F 𝑇𝑥=𝑥 1 𝑒 𝑖2𝜋 𝑓 1 𝑡 + 𝑥 2 𝑒 𝑖2𝜋 𝑓 2 𝑡 + 𝑥 3 𝑒 𝑖2𝜋 𝑓 3 𝑡 + 𝑥 4 𝑒 𝑖2𝜋 𝑓 4 𝑡 𝑓 4 𝑓 3 𝑓 2 𝑓 1 𝑓 1 =0, 𝑓 2 = 𝑓 𝑠 𝑁 , 𝑓 3 = 2 𝑓 𝑠 𝑁 ……. 𝑓 𝑁 = 𝑁−1 𝑓 𝑠 𝑁 𝑇𝑥= 𝑥 1 +𝑥 2 + 𝑥 3 + .. + 𝑥 𝑁−1 𝑒 𝑖2𝜋 𝑁 𝑒 𝑖2∗2𝜋 𝑁 .. 𝑒 𝑖(𝑁−1)2𝜋 𝑁 𝑒 𝑖2∗2∗2𝜋 𝑁 𝑒 𝑖 𝑁−1 ∗2∗2𝜋 𝑁 𝑒 𝑖(𝑁−1)∗2𝜋 𝑁 𝑒 𝑖2∗(𝑁−1)∗2𝜋 𝑁 𝑒 𝑖 𝑁−1 ∗(𝑁−1)∗2𝜋 𝑁 1

𝑇𝑥= 𝑥 1 +𝑥 2 + 𝑥 3 + .. + 𝑥 𝑁−1 𝑻𝒙=𝑰𝑭𝑭𝑻( 𝒙 𝟏 , 𝒙 𝟐 …. , 𝒙 𝒊 … 𝒙 𝒏 ) 𝑇𝑥= 𝑥 1 +𝑥 2 + 𝑥 3 + .. + 𝑥 𝑁−1 𝑒 𝑖2𝜋 𝑁 𝑒 𝑖2∗2𝜋 𝑁 .. 𝑒 𝑖(𝑁−1)2𝜋 𝑁 𝑒 𝑖2∗2∗2𝜋 𝑁 𝑒 𝑖 𝑁−1 ∗2∗2𝜋 𝑁 𝑒 𝑖(𝑁−1)∗2𝜋 𝑁 𝑒 𝑖2∗(𝑁−1)∗2𝜋 𝑁 𝑒 𝑖 𝑁−1 ∗(𝑁−1)∗2𝜋 𝑁 1 𝑒 𝑖2𝜋 𝑁 𝑒 𝑖2∗2𝜋 𝑁 .. 𝑒 𝑖(𝑁−1)2𝜋 𝑁 𝑒 𝑖2∗2∗2𝜋 𝑁 𝑒 𝑖 𝑁−1 ∗2∗2𝜋 𝑁 𝑒 𝑖(𝑁−1)∗2𝜋 𝑁 𝑒 𝑖2∗(𝑁−1)∗2𝜋 𝑁 𝑒 𝑖 𝑁−1 ∗(𝑁−1)∗2𝜋 𝑁 𝑥 1 𝑥 2 𝑥 𝑁 𝑇𝑥= 1 𝑻𝒙=𝑰𝑭𝑭𝑻( 𝒙 𝟏 , 𝒙 𝟐 …. , 𝒙 𝒊 … 𝒙 𝒏 )

OFDM (Orthogonal frequency division multiplexing) transmission sin⁡(2𝜋𝑓𝑡) x cos⁡(2𝜋𝑓𝑡) + I = Real part Q = Imag part 𝑓 1 𝑓 2 𝑓 3 𝑓 4 𝑓 5 IFFT 𝑓 4 𝑓 3 𝑓 2 𝑓 1 𝑓 5 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑥 5 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑥 5 −𝑓 𝑓

OFDM performance under deep fading 𝑋 𝑓 𝐹 𝑐 =−𝑓 𝐹 𝑐 =𝑓 𝐻 𝑓 𝐹 𝑐 =−𝑓 𝐹 𝑐 =𝑓 𝐹 𝑐 =𝑓 𝐹 𝑐 =−𝑓

OFDM reception x 𝑓 1 𝑓 2 𝑓 3 𝑓 4 𝑓 5 𝑠𝑖𝑛⁡(2𝜋𝑓𝑡) cos⁡(2𝜋𝑓𝑡) 𝑅𝑒 𝑚 = 𝐼(𝑡) 𝐼𝑚 𝑚 = 𝑄(𝑡) 𝐹𝐹𝑇 𝑓 1 𝑓 2 𝑓 3 𝑓 4 𝑓 5 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑥 5 𝐵𝑎𝑠𝑒𝑏𝑎𝑛𝑑 𝑓 4 𝑓 3 𝑓 2 𝑓 1 𝑓 5 OFDM receives messages proportional to the number of good frequencies (not faded), instead of losing out the whole message

OFDM vs Conventional Robust to deep fading Very efficient, achieves capacity limits, used widely in LTE/WiFi Robust to synchronization errors Requires FFT/IFFT power intensive High variation in signal amplitude – needs better h/w Complete loss of performance under deep fading Cannot reach maximum capacity High synchronization overhead Suitable for low power/battery- less communication Low variation in signal amplitude