Modulation and OFDM

Communication Exchange of information from point A to point B Transmit Receive 100001101010001011101 100001101010001011101

Wireless Communication Exchange of information from point A to point B Transmit Receive 100001101010001011101 100001101010001011101

Wireless Communication Exchange of information from point A to point B Modulation Upconvert 100001101010001011101 Downconvert Demodulation 100001101010001011101

Modulation Converting bits to signals These signals are later sent over the air The receiver picks these signals and decodes transmitted data Modulation 100001101010001011101 Signals (voltages)

Amplitude Modulation Suppose we have 4 voltage levels (analog) to represent bits. 00 01 10 11 𝑉 4 - 𝑉 4 3𝑉 4 − 3𝑉 4

Amplitude Modulation - 𝑉 4 - 𝑉 4 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 00 3𝑉 4 𝑉 4 01 - 𝑉 4 10 3𝑉 4 11 − 3𝑉 4 𝑉 4 - 𝑉 4 Individual voltage levels are called as symbols − 3𝑉 4

Modulated symbols for transmission 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 FFT -F Frequency F

Received symbols with distortions 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 FFT -F Frequency F

Demodulation Tx bits 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 00 3𝑉 4 𝑉 4 01 - 𝑉 4 10 3𝑉 4 11 − 3𝑉 4 𝑉 4 - 𝑉 4 − 3𝑉 4 0 0 0 0 1 0 Rx bits decoded 1 0 1 0 1 0 0 0 1 1 1 1 1 1

Coping up with demodulation errors If the noise is too high, there may be too many bit flips Symbols for modulation to be chosen as a function of this noise For example, if we want to eliminate bit flips completely, we can choose voltage levels as follows

Modulation with sparser symbols 1 𝑉 −𝑉 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 𝑉 −𝑉

Received symbols with distortion 1 𝑉 −𝑉 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 𝑉 −𝑉

Demodulation 1 𝑉 −𝑉 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 𝑉 −𝑉 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0

That eliminated all the bit flips, which is good However, what is the disadvantage of choosing only two voltage levels? Takes longer to transmit, hence bit rate is very low

Bit rates Bit per symbol 𝑛𝑏 𝑠 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 𝐵 (bandwidth) Symbol rate (Baud rate) R s = 1 𝑡 𝑠 Frequency Symbol duration 𝑡 𝑠 -F F 𝑉 FFT 𝑉 2 Bit rate 𝑅 𝑏 =𝑛 𝑏 𝑠 ∗ 𝑅 𝑠 - 𝑉 2 Symbol rate (Baud rate) R b =2∗𝐵 (bandwidth) Bit rate 𝑅 𝑏 =2∗𝑛 𝑏 𝑠 ∗𝐵 −𝑉

Transmission of modulated symbols The modulated message has zero center frequency (baseband) Impractical to have antennas at that frequencies Causes interference if everyone wants to use baseband .. 𝐵 (bandwidth) 𝐹 𝑐 =2.4𝐺𝐻𝑧 𝐹 𝑐 =−2.4𝐺𝐻𝑧 Upconversion 𝐵 (bandwidth) 𝑐𝑒𝑛𝑡𝑒𝑟 𝑓𝑟𝑒𝑞 𝐹 𝑐 =0

Upconversion  shifting center frequency 𝑓 −𝑓 cos⁡(2𝜋𝑓𝑡)= 𝑚 𝑡 = 𝐵 (bandwidth) 𝐹 𝑐 =0 𝐵 (bandwidth) 𝐹 𝑐 =𝑓 𝐹 𝑐 =−𝑓 𝑢 𝑡 =𝑚 𝑡 cos 2𝜋𝑓𝑡 =

Down-conversion  bringing signal back to baseband The receiver needs to perform an operation of down-conversion The received signal is a high frequency signal in RF Processing the data at these frequencies needs high clock digital circuits, which is impractical We need to convert the data back to baseband and process the low frequency signals for decoding bits

Down-conversion  bringing signal back to baseband 𝑢 𝑡 =𝑚 𝑡 cos 2𝜋𝑓𝑡 = 𝐵 (bandwidth) 𝐹 𝑐 =−𝑓 𝐹 𝑐 =𝑓 Low pass filter eliminates this Then, we recover baseband signal 𝐵 (bandwidth) 𝐹 𝑐 =2𝑓 𝐹 𝑐 =0 d 𝑡 =𝑢 𝑡 cos 2𝜋𝑓𝑡 = 𝑚 𝑡 cos 2 (2𝜋𝑓𝑡) d 𝑡 =𝑚 𝑡 (1+𝑐𝑜𝑠 (4𝜋𝑓𝑡) ) 2 d 𝑡 = 𝑚(𝑡) 2 + 𝑐𝑜𝑠 (4𝜋𝑓𝑡) 2

Upconversion and Downconversion summary 𝑢(𝑡) 2 + 𝑐𝑜𝑠 (4𝜋𝑓𝑡) 2 m(t) r(t) x x cos⁡(2𝜋𝑓𝑡) cos⁡(2𝜋𝑓𝑡)

Upconversion and Downconversion summary 𝑢(𝑡) 2 + 𝑐𝑜𝑠 (4𝜋𝑓𝑡) 2 I(t) r(t) x x cos⁡(2𝜋𝑓𝑡) cos⁡(2𝜋𝑓𝑡)

Beyond amplitude modulation We have learnt communication with amplitude modulation There is a simple idea to double the data rate using QAM (quadrature amplitude modulation)

Quadrature amplitude modulation Achieves double data rate compared to amplitude modulation alone x cos⁡(2𝜋𝑓𝑡) x cos⁡(2𝜋𝑓𝑡) I(t) + 𝐼 𝑡 2 + 𝐼 𝑡 cos 4𝜋𝑓𝑡 + 𝑄 𝑡 𝑠𝑖𝑛 4𝜋𝑓𝑡 2 Q(t) x 𝑠𝑖𝑛⁡(2𝜋𝑓𝑡) x sin⁡(2𝜋𝑓𝑡) 𝐼(𝑡)𝑐𝑜𝑠 2𝜋𝑓𝑡 +𝑄(𝑡)𝑠𝑖𝑛(2𝜋𝑓𝑡) 𝑄 𝑡 2 + 𝐼 𝑡 cos 4𝜋𝑓𝑡 + 𝑄 𝑡 𝑠𝑖𝑛 4𝜋𝑓𝑡 2

This scheme uses 16 symbols (4 bits per symbol), hence called 16 QAM Symbols with QAM 𝐼 0010 0011 0001 0000 0110 0111 0100 1110 1111 1101 1100 1010 1011 1001 1000 3𝑉 4 𝑉 4 3𝑉 4 - 3𝑉 4 − 𝑉 4 𝑉 4 𝑄 − 𝑉 4 − 3𝑉 4 This scheme uses 16 symbols (4 bits per symbol), hence called 16 QAM

64 QAM Denser modulation can be used when symbol distortion is less in the channel

BPSK (binary phase shift keying) Coarser modulation can be used when symbol distortion is huge

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

Deep channel fading 𝑋 𝑓 𝐻 𝑓 𝐹 𝑐 =−𝑓 𝐹 𝑐 =𝑓 𝐹 𝑐 =−𝑓 𝐹 𝑐 =𝑓 𝑌=𝐻∗𝑋 𝑌 𝑓 = 𝐻 𝑓 𝑋 𝑓

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

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

+ + + 𝑒 𝑖2𝜋 𝑓 1 𝑡 𝑒 𝑖2𝜋 𝑓 2 𝑡 𝑒 𝑖2𝜋 𝑓 3 𝑡 𝑒 𝑖2𝜋 𝑓 4 𝑡 x x x x F -F 𝑥(𝑡) 𝑥 1 (𝑡) 𝑥 1 𝑥 2 𝑥 3 𝑥 4 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 𝑥 4 (𝑡) 𝑒 𝑖2𝜋 𝑓 4 𝑡

𝑇𝑥=𝐼𝐹𝐹𝑇( 𝑥 1 𝑡 1 , 𝑥 2 (𝑡) …. , 𝑥 𝑖 (𝑡) … 𝑥 𝑛 (𝑡)) F -F 𝑥 1 𝑥 2 𝑥 3 𝑥 4 𝑇𝑥=𝑥 1 (𝑡) 𝑒 𝑖2𝜋 𝑓 1 𝑡 + 𝑥 2 (𝑡) 𝑒 𝑖2𝜋 𝑓 2 𝑡 + 𝑥 3 (𝑡) 𝑒 𝑖2𝜋 𝑓 3 𝑡 + 𝑥 4 (𝑡) 𝑒 𝑖2𝜋 𝑓 4 𝑡 𝑇𝑥= ∑ 𝑥 𝑖 (𝑡)𝑒 𝑖2𝜋 𝑓 𝑖 𝑡 𝑓 4 𝑓 3 𝑓 2 𝑓 1 𝑇𝑥=𝐼𝐹𝐹𝑇( 𝑥 1 𝑡 1 , 𝑥 2 (𝑡) …. , 𝑥 𝑖 (𝑡) … 𝑥 𝑛 (𝑡))

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 cos⁡(2𝜋𝑓𝑡) 𝑅𝑒 𝑚 = 𝐼(𝑡) 𝐼𝑚 𝑚 = 𝑄(𝑡) 𝑥 1 𝑓 1 𝐵𝑎𝑠𝑒𝑏𝑎𝑛𝑑 𝑓 4 𝑓 3 𝑓 2 𝑓 1 𝑓 5 𝑓 2 𝑥 2 𝑓 3 𝑥 3 𝐹𝐹𝑇 𝑓 4 𝑥 4 𝑓 5 𝑥 5 𝑠𝑖𝑛⁡(2𝜋𝑓𝑡)

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