Modulation and Demodulation in Mobile Communication

Orthogonal Frequency Division Multiplexing (OFDM) is a multicarrier modulation in which a high-rate data stream is split into N low-rate streams, and each is sent on a separate subcarrier. The subcarriers are spaced at Δf = 1/T_s (T_s = OFDM symbol time), so they are orthogonal and their spectra can overlap without interference. Because each subcarrier carries a slow symbol, the channel looks flat on each subcarrier, which removes the need for a complex equalizer. It is used in Wi-Fi, LTE, 5G NR, DVB-T and WiMAX.

OFDM transmitter

bits → [Channel  ] → [Symbol  ] → [S/P] → [IFFT ]
       [coding + ]   [mapping ]           [N-pt ]
       [interleave]  [QAM/PSK ]              |
                                              v
 RF ← [Up-conv/] ← [DAC] ← [P/S] ← [Add cyclic ]
      [  PA    ]                    [ prefix   ]

1. Coding and interleaving – adds FEC and spreads burst errors.
2. Symbol mapping – groups bits into QPSK/16-QAM/64-QAM symbols X_k.
3. Serial-to-parallel (S/P) – splits the symbol stream into N parallel streams, one per subcarrier; pilots are also inserted.
4. IFFT – N-point IFFT does the multicarrier modulation in one step: x(n) = (1/N) Σ X_k e^(j2πkn/N). It turns frequency-domain symbols into a time-domain OFDM symbol.
5. Parallel-to-serial (P/S) – forms one time sequence.
6. Cyclic prefix (CP) – the last part of the symbol is copied to its front. CP length is greater than the channel delay spread, so it absorbs ISI and keeps subcarriers orthogonal.
7. DAC, up-conversion and power amplifier – produce the RF signal.

OFDM receiver

RF → [Down-conv] → [ADC] → [Remove CP] → [S/P]
                                           |
                                           v
 bits ← [Decode/  ] ← [Demap] ← [P/S] ← [Equalize] ← [FFT]
        [deinterl.]                     [1-tap  ]

1. Down-conversion and ADC – baseband samples; time and frequency synchronization are done here.
2. Remove CP – the prefix, which holds the ISI, is discarded.
3. S/P and FFT – the N-point FFT separates the subcarriers: Y_k = H_k X_k + noise.
4. One-tap equalizer – each subcarrier is divided by its channel estimate H_k (found from pilots).
5. P/S, demapping, deinterleaving and decoding – recover the original bits.

Key points

• Long symbol duration + CP gives strong resistance to multipath and ISI.
• FFT/IFFT make the system cheap to build.
• Drawbacks: high peak-to-average power ratio (PAPR) and sensitivity to frequency offset and Doppler.

Quadrature Phase Shift Keying (QPSK) is a 4-level PSK in which each symbol carries 2 bits, and the carrier takes one of four phases (π/4, 3π/4, 5π/4, 7π/4). It needs the same bandwidth as BPSK for half the data rate, so it has twice the bandwidth efficiency (2 bps/Hz ideally) with the same bit error rate.

s_i(t) = √(2E_s/T_s) cos(2π f_c t + (i − 1)π/2),  i = 1..4

It can be written as two BPSK signals in quadrature: an in-phase (I) part on cos and a quadrature (Q) part on sin.

Constellation diagram

              Q
              |
     01  •    |    •  00
              |
   -----------+-----------  I
              |
     11  •    |    •  10
              |

Gray coding is used, so neighbouring points differ by only one bit. Distance between neighbouring points = √(2E_s).

QPSK transmitter

           +-> [I: odd bits ] -> (x) --+
           |   [NRZ, Rb/2   ]    ^     |
bits ->[S/P]                cos ωct    (+)-> QPSK
 (Rb)      |                     v     |
           +-> [Q: even bits] -> (x) --+
               [NRZ, Rb/2   ]  sin ωct

1. The unipolar data at rate R_b is converted to bipolar NRZ (±1).
2. A serial-to-parallel converter splits it into the I stream (odd bits) and Q stream (even bits), each at R_b/2.
3. Each stream may pass a pulse-shaping filter (e.g. raised cosine) to limit bandwidth.
4. The I stream multiplies cos(ω_c t) and the Q stream multiplies sin(ω_c t); the two BPSK signals are added to give QPSK.

QPSK receiver (coherent detection)

          +-> (x) -> [LPF/Integ] -> [Decision] -+
          |    cos ωct                          |
QPSK -> [BPF]                                [P/S]-> bits
          |    sin ωct                          |
          +-> (x) -> [LPF/Integ] -> [Decision] -+
               ^
         [Carrier recovery]

1. The input passes a band-pass filter to remove out-of-band noise.
2. A carrier recovery circuit produces coherent cos and sin references.
3. Two correlators (multiplier + integrator over T_s) recover the I and Q components.
4. Decision circuits compare each output with zero to decide 1 or 0.
5. A parallel-to-serial converter combines the two streams into the original bit stream.

Properties

• Bit error rate: P_b = Q(√(2E_b/N₀)), same as BPSK.
• Phase changes of up to 180° can occur, which causes envelope dips after filtering; OQPSK and π/4-QPSK reduce this.
• Used in satellite links, CDMA (IS-95), WCDMA and LTE.

Spread spectrum modulation

Spread spectrum (SS) modulation is a technique in which the transmitted signal occupies a bandwidth much larger than the minimum needed for the data. The spreading is done by a pseudo-noise (PN) code that is independent of the data, and the receiver uses the same synchronized code to despread it. The ratio of spread bandwidth to data rate is the processing gain PG = B_ss/R_b = T_b/T_c. SS gives resistance to jamming, interference and multipath, low probability of interception, and allows multiple access (CDMA).

Direct sequence spread spectrum (DSSS)

In DSSS, each data bit is multiplied directly by a fast PN sequence with chip period T_c ≪ T_b. The product is then BPSK modulated on the carrier.

s(t) = √(2E_s/T_s) m(t) p(t) cos(2π f_c t + θ)

where m(t) is the data and p(t) the PN code.

Transmitter

data m(t) ---> (x) ------> (x) ------> s(t)
  Rb            ^           ^        (wideband)
                |           |
          [PN generator] [Oscillator]
           (chip Rc)      cos ωct

1. Data m(t) (±1, rate R_b) is multiplied by the PN code p(t) (±1, rate R_c = N R_b).
2. The product is a wideband signal of bandwidth ≈ 2R_c.
3. It is BPSK modulated onto the carrier and transmitted.

Receiver

r(t)->[BPF]->(x)--->(x)--->[Integrate]->[Decision]-> m^(t)
              ^      ^      [0..Tb    ]
              |      |
       [PN generator] [Carrier]
       [+ sync     ]  [recovery]

1. The received signal r(t) = s(t) + interference + noise is filtered.
2. It is multiplied by a locally generated, synchronized copy of p(t). Because p(t)·p(t) = 1, the wanted signal collapses back to its narrow data bandwidth (despreading).
3. Any narrowband interference gets multiplied by p(t) and is spread over the wide band, so most of it is removed by the following filter.
4. Coherent demodulation (multiply by carrier, integrate over T_b, decide) recovers the data.

 Spectrum at receiver:
 before despreading     after despreading
   jammer                 data
     |                     /\
  ___|___               __/  \__   jammer spread
 /  data \___         ___________  (low level)

Example: IS-95 CDMA uses a chip rate of 1.2288 Mcps for 9.6 kbps data, giving PG = 128 (21 dB). IEEE 802.11b also uses DSSS.

OFDM transmitting side

OFDM sends a high-rate data stream as many low-rate streams on closely spaced orthogonal subcarriers (spacing Δf = 1/T_s). The transmitter does this with an IFFT.

bits->[FEC +     ]->[QAM/PSK]->[S/P]->[  IFFT  ]
      [interleave]  [mapper ]         [N-point ]
                                          |
RF <-[Up-conv]<-[DAC]<-[P/S]<-[Add CP]<---+
     [+ PA   ]

1. Channel coding and interleaving: adds redundancy (e.g. convolutional code) and spreads bits across subcarriers so one deep fade does not wipe out consecutive bits.
2. Mapping: bits are grouped and mapped to complex symbols (BPSK, QPSK, 16-QAM, 64-QAM).
3. Serial-to-parallel: N symbols are placed on N subcarriers; pilot tones and null (guard) carriers are added.
4. IFFT: computes x(n) = (1/N) Σ_{k=0}^{N−1} X_k e^(j2πkn/N), i.e. modulates every subcarrier at once and produces one time-domain OFDM symbol.
5. Parallel-to-serial and cyclic prefix: the last N_cp samples are copied to the front. If the CP is longer than the maximum delay spread, ISI is absorbed and orthogonality is kept.
6. DAC, filter, up-converter and power amplifier: send the signal at RF.

Advantages

• Strong resistance to multipath fading and ISI (long symbols + CP).
• Simple one-tap equalization per subcarrier instead of a long time-domain equalizer.
• High spectral efficiency because subcarrier spectra overlap.
• Efficient implementation with FFT/IFFT.
• Flexible: bit loading and different modulation per subcarrier; easy to extend to OFDMA and MIMO.
• Robust against narrowband interference (only a few subcarriers affected).

Disadvantages

• High peak-to-average power ratio (PAPR), needing linear, less efficient power amplifiers.
• Very sensitive to carrier frequency offset and Doppler, which destroy orthogonality (inter-carrier interference).
• Cyclic prefix and pilots waste some bandwidth and power.
• Needs accurate time and frequency synchronization.

Gaussian Minimum Shift Keying (GMSK) is MSK in which the NRZ data is first passed through a Gaussian low-pass pre-modulation filter before frequency modulation. The filter smooths the phase path, which suppresses side lobes and gives a very compact spectrum, while the envelope stays constant. GMSK with BT = 0.3 is the modulation used in GSM.

The Gaussian filter impulse response is:

h_G(t) = (√π/α) exp(−π² t² / α²),  α = 0.5887 / B

where B is the 3 dB bandwidth of the filter. The product BT (B × bit period) controls the trade-off: smaller BT gives a narrower spectrum but more ISI. BT = ∞ is plain MSK.

GMSK transmitter

Method 1 – direct FM:

NRZ data -> [Gaussian LPF] -> [FM / VCO     ] -> GMSK
            (BT = 0.3)        [h = 0.5      ]

The filtered data drives a voltage-controlled oscillator with modulation index h = 0.5. Simple, but the VCO must hold the exact index.

Method 2 – quadrature (I/Q) modulator:

data->[Gaussian LPF]->[Integrator]-> φ(t)
                                     |
              +----------------------+
              v                      v
           [cos φ]->(x) cos ωct   [sin φ]->(x) sin ωct
                     |                      |
                     +--------(+)-----------+
                               |
                             GMSK

The phase φ(t) is generated digitally; cos φ and sin φ modulate the I and Q carriers. This keeps the index exact and is used in practice.

GMSK receiver (coherent)

       +->(x)->[LPF]->[Sample]->I -+
       |  cos ωct                  |
GMSK->-+                       [Decision / ]-> data
       |  sin ωct              [ Viterbi   ]
       +->(x)->[LPF]->[Sample]->Q -+

The signal is multiplied by cos and sin of the recovered carrier, low-pass filtered and sampled; I and Q are sampled at alternate bit instants (like MSK/OQPSK detection). Because of controlled ISI, GSM uses a Viterbi (MLSE) equalizer. GMSK can also be detected non-coherently with a frequency discriminator.

Constellation / phase diagram

              Q
              • (π/2)
              |
   (π) •------+------• (0)   I
              |
              • (3π/2)
 Constant envelope: points on a circle;
 phase moves ±π/2 per bit along smooth arcs.

Features

• Constant envelope, so efficient non-linear class-C amplifiers can be used (good battery life).
• Very narrow spectrum, low adjacent channel interference (GSM: 270.833 kbps in 200 kHz).
• BER is slightly worse than MSK due to ISI from the filter.

Direct Sequence Spread Spectrum (DSSS) spreads a narrowband data signal over a wide bandwidth by multiplying each data bit with a fast pseudo-noise (PN) code whose chip period T_c is much smaller than the bit period T_b. The receiver multiplies again by the same synchronized code to recover the data.

s(t) = A m(t) p(t) cos(2π f_c t)
Processing gain PG = T_b / T_c = R_c / R_b

Transmitter

m(t) ±1 --->(x)--- m(t)p(t) --->(x)---> s(t)
  (Rb)       ^                  ^
             |                  |
       [PN code p(t)]     [cos ωct]
       [ rate Rc    ]

• Data m(t) (±1) is multiplied by PN code p(t) (±1). Each bit becomes N = T_b/T_c chips.
• The chip stream BPSK-modulates the carrier. Bandwidth grows from about 2R_b to 2R_c.

Receiver

r(t)->(x)------>(x)---->[∫ over Tb]->[Decision]->m^(t)
       ^         ^
  [Local PN ] [cos ωct]
  [synced   ]

• The received signal is multiplied by a locally generated PN code, aligned by an acquisition and tracking circuit.
• Since p(t)² = 1, the wanted signal is despread to its original data bandwidth.
• Narrowband interference or jamming is spread by the same multiplication and mostly filtered out.
• Coherent BPSK demodulation (multiply, integrate over T_b, threshold) gives the data.

Waveforms (one data bit = 7 chips)

m(t):   ‾‾‾‾‾‾‾‾‾‾‾‾‾‾|______________
p(t):   ‾|_|‾‾|_|‾|_‾ |‾|_|‾‾|_|‾|_
m·p :   ‾|_|‾‾|_|‾|_‾ |_|‾|__|‾|_|‾

Features

• Interference suppression by a factor equal to PG.
• Multipath rejection: delayed copies more than one chip late look like noise; a RAKE receiver can combine them.
• Allows CDMA: users share the band using different codes. Used in IS-95, WCDMA and IEEE 802.11b.

MSK as continuous phase FSK

Minimum Shift Keying (MSK) is a binary continuous phase FSK (CPFSK) with modulation index h = 0.5, the minimum index for which the two tones are orthogonal over one bit (coherent detection).

s(t) = √(2E_b/T_b) cos(2π f_c t + θ(t)),
θ(t) = ± π t / (2T_b) + θ(0)
f_H − f_L = 1 / (2T_b) = R_b / 2

The phase changes linearly by exactly ±π/2 in each bit and has no jumps, so the waveform is continuous and the spectrum falls off fast.

MSK as a special OQPSK

MSK can also be written as OQPSK with half-sinusoidal pulse shaping instead of rectangular pulses:

s(t) = a_I(t) cos(πt/2T_b) cos(2π f_c t)
     + a_Q(t) sin(πt/2T_b) sin(2π f_c t)

• a_I and a_Q are the even and odd bits, each lasting 2T_b.
• The Q stream is offset by T_b from I (as in OQPSK).
• The half-sine weighting makes the envelope constant and the phase continuous.

Comparison

Notes on the table: in null-to-null terms, OQPSK main lobe width is R_b (narrower than MSK's 1.5R_b), but its side lobes fall slowly (1/f²); MSK side lobes fall as 1/f⁴; GMSK shrinks side lobes further.

Suitable for GSM

GMSK is used in GSM because:

• Its constant envelope lets mobiles use non-linear class-C amplifiers, which are power efficient and save battery.
• Its narrow spectrum and low side lobes give low adjacent channel interference, allowing 270.833 kbps in a 200 kHz channel.
• It can be detected coherently or non-coherently, and the small ISI is handled by GSM's Viterbi equalizer.

Advantages of digital over analog modulation

• Better noise and fading immunity; signals can be regenerated.
• Supports error detection/correction coding, encryption and source coding.
• Easy multiplexing of voice, data and video (TDMA, CDMA, OFDM).
• Higher spectral and power efficiency, and cheaper VLSI/DSP implementation.

GMSK modulation

Gaussian MSK is MSK in which the bipolar NRZ data passes through a Gaussian low-pass filter before frequency modulation with index h = 0.5. The filter smooths the phase trajectory, which reduces side lobes and out-of-band power. The 3 dB bandwidth–bit period product BT controls the shape: GSM uses BT = 0.3, and BT = ∞ gives MSK.

h_G(t) = (√π/α) exp(−π² t² / α²),  α = 0.5887 / B

Transmitter

NRZ -> [Gaussian LPF] -> [Integrator] -> φ(t)
data   (BT = 0.3)                        |
                    +---------+----------+
                    v                    v
                  [cos φ]              [sin φ]
                    |                    |
           cos ωct->(x)          sin ωct->(x)
                    +------(+)-----------+
                            |
                          GMSK

• The Gaussian filter spreads each bit over about 3 bit periods (controlled ISI).
• The filtered signal is integrated to give the phase φ(t); I = cos φ, Q = sin φ modulate quadrature carriers. (A simpler form feeds the filtered data directly to a VCO with h = 0.5.)

Receiver

GMSK->[BPF]-+->(x)->[LPF]->[Sample]->I-+
            |  cos                     |
            |                    [MLSE / ]->data
            +->(x)->[LPF]->[Sample]->Q-+ [decision]
               sin    ^
                 [Carrier & clock recovery]

• Coherent I/Q demodulation recovers the phase; I and Q are sampled at alternate bit instants as in MSK.
• Because the Gaussian filter adds ISI, GSM receivers use a Viterbi (MLSE) equalizer. Non-coherent detection with a limiter–discriminator is also possible.

Constellation (phase states)

            Q
            •
            |
     •------+------•  I
            |
            •
 Constant envelope; phase moves
 smoothly by ±π/2 per bit.

Merits: constant envelope (class-C amplifiers, long battery life), narrow spectrum (low adjacent channel interference), used in GSM, DECT and Bluetooth (GFSK variant).

MSK modulation

Minimum Shift Keying (MSK) is a continuous phase binary FSK with modulation index h = 0.5, i.e. tone spacing Δf = R_b/2, the minimum spacing that keeps the two tones orthogonal. The phase changes linearly by ±π/2 per bit with no sudden jumps, so the envelope is constant. It can also be seen as OQPSK with half-sinusoidal pulse shaping. MSK has fast side-lobe roll-off and the same BER as BPSK.

Direct sequence spread spectrum (DSSS)

DSSS spreads the data over a wide band by multiplying each data bit with a fast PN code of chip period T_c ≪ T_b. Processing gain PG = T_b/T_c.

Transmitter

m(t) --->(x)--->(x)---> s(t) = m(t)p(t)cos ωct
          ^      ^
    [PN gen p(t)] [cos ωct]

• Data (±1) × PN code (±1) gives a chip stream at rate R_c.
• The chip stream BPSK-modulates the carrier; the bandwidth becomes ≈ 2R_c.

Receiver

r(t)->(x)--->(x)--->[∫ 0..Tb]->[Decision]-> m^(t)
       ^      ^
 [Local PN  ] [Carrier]
 [synchron. ] [recovery]

• Multiplying by the same synchronized PN code despreads the wanted signal (p² = 1).
• Interference is spread by the same step and mostly filtered out.
• Integrating over T_b and comparing with zero gives the data.

 Before despread      After despread
   ____________          /\   <- data
  |  wideband  |    ____/  \____ noise

DSSS gives anti-jam capability, multipath resistance (RAKE) and CDMA multiple access; it is used in IS-95, WCDMA and IEEE 802.11b.

The OFDM receiver reverses the steps of the transmitter. It uses an FFT to separate the orthogonal subcarriers, after which each subcarrier sees a flat channel and is equalized with one complex gain.

RF->[Down-conv]->[ADC]->[Sync]->[Remove]->[S/P]
                                 [ CP   ]    |
                                             v
bits<-[Decode]<-[Demap]<-[P/S]<-[1-tap EQ]<-[FFT]
                                   ^
                            [Channel est.]
                            [from pilots ]

1. Down-conversion and ADC – RF to complex baseband samples.
2. Synchronization – finds the symbol start and corrects carrier frequency offset (often using the CP or preambles).
3. Remove cyclic prefix – discards the guard part that contains ISI from the previous symbol.
4. S/P and N-point FFT – converts the time samples back to subcarrier symbols: Y_k = H_k X_k + N_k.
5. Channel estimation and equalization – H_k is estimated from pilots and X̂_k = Y_k / H_k (zero-forcing) per subcarrier.
6. P/S, demapping, deinterleaving and decoding – recover the bit stream.

Because equalization is one division per subcarrier, the receiver is far simpler than a time-domain equalizer for the same multipath channel.

Spread spectrum

Spread spectrum (SS) is a modulation technique in which the signal is spread over a bandwidth much larger than the data rate needs, using a pseudo-noise (PN) code that is independent of the data. The receiver uses the same synchronized code to despread it. It gives anti-jamming capability, low probability of interception, multipath resistance and multiple access (CDMA). Processing gain PG = B_ss / R_b.

Frequency hopping spread spectrum (FHSS)

In FHSS, the carrier frequency of a narrowband signal (usually M-ary FSK) jumps randomly among a large set of channels. The hopping pattern is decided by a PN code known to both transmitter and receiver. At any instant the signal occupies only one narrow channel, but over time it covers the whole spread band.

Transmitter

data->[FSK    ]--->(x)--->[BPF]---> FH signal
      [modulator]   ^
                    |
            [Frequency synthesizer]
                    ^
             [PN code generator]

Receiver

FH in-->(x)-->[BPF]-->[FSK      ]--> data
         ^            [demodulator]
         |
 [Frequency synthesizer]
         ^
 [PN generator + sync  ]

• At the transmitter, the PN generator selects a frequency from the synthesizer; mixing shifts the FSK signal to that hop channel.
• At the receiver, an identical synchronized synthesizer "de-hops" the signal back to a fixed IF, and a normal FSK demodulator recovers the data.

Time–frequency pattern

 f
 |      ■           ■
 |  ■         ■
 |        ■             ■
 |    ■           ■
 +------------------------> t
   one hop per dwell time

Types

• Slow hopping: several symbols are sent per hop (hop rate < symbol rate).
• Fast hopping: one symbol is spread over several hops (hop rate > symbol rate); better against fading and jamming.

Features: A jammer or deep fade affects only a few hops, which coding and interleaving can correct. FHSS does not need the very fast chip-level synchronization of DSSS and avoids the near-far problem. It is used in Bluetooth (1600 hops/s over 79 channels) and optionally in GSM (slow frequency hopping).

The OFDM transmitter divides a high-rate data stream into N low-rate streams and places them on N orthogonal subcarriers (spacing Δf = 1/T_s) using an IFFT, then adds a cyclic prefix to fight multipath.

bits->[FEC +]->[QAM  ]->[S/P]->[IFFT ]->[P/S]
      [intlv]  [map  ]          [N-pt ]    |
                                            v
RF <-[Up-conv + PA]<-[DAC]<-[Add cyclic prefix]

1. Coding and interleaving – add error-correction redundancy and spread bits across subcarriers.
2. Symbol mapping – map bits to QPSK/16-QAM/64-QAM symbols X_k.
3. Serial-to-parallel – N symbols assigned to N subcarriers; pilots inserted.
4. IFFT – x(n) = (1/N) Σ X_k e^(j2πkn/N) modulates all subcarriers at once into one time-domain OFDM symbol.
5. Parallel-to-serial and cyclic prefix – the last few samples are copied to the front; CP longer than the delay spread prevents ISI.
6. DAC, up-conversion and power amplifier – produce the RF signal.

The IFFT replaces a bank of N oscillators, making OFDM cheap to implement in Wi-Fi, LTE and DVB-T.

Need for M-ary QAM

In M-ary PSK all points lie on one circle, so as M increases the points crowd together and the noise margin falls quickly. M-ary QAM varies both amplitude and phase, spreading the points over the I–Q plane. This gives:

• High bandwidth efficiency: log₂M bits per symbol (16-QAM: 4 bits, 64-QAM: 6 bits) in the same bandwidth.
• Better noise performance than M-PSK for the same M and same average power, because the minimum distance is larger.
• Support for high data rates in limited spectrum (LTE, Wi-Fi, DVB, cable modems) and adaptive modulation.

Constellation diagrams

     16-QAM (square)          16-PSK (circle)
         Q                         Q
  •   •  |  •   •               •  •  •
  •   •  |  •   •            •     |     •
---------+---------  I     •       |       •
  •   •  |  •   •          •-------+-------•  I
  •   •  |  •   •          •       |       •
                             •     |     •
                                •  •  •
 levels ±1, ±3 on I and Q   16 points, 22.5° apart

Minimum distance (same average symbol energy E_s)

16-PSK: d_min = 2√E_s sin(π/16) = 0.390 √E_s
16-QAM: d_min = 2√(E_s/10)      = 0.632 √E_s
Ratio  = 20 log(0.632/0.390)    ≈ 4.2 dB

So 16-QAM needs about 4.2 dB less power than 16-PSK for the same error rate.

Comparison

Both carry 4 bits per symbol, but 16-QAM uses the signal space more efficiently, which is why QAM is preferred for M ≥ 16.

MSK

Minimum Shift Keying (MSK) is binary continuous phase FSK with modulation index h = 0.5, so the tone spacing is R_b/2, the minimum for orthogonal coherent detection.

• The phase changes linearly by ±π/2 in each bit, with no discontinuity.
• Equivalent to OQPSK with half-sinusoidal pulse shaping: s(t) = a_I cos(πt/2T_b) cos ω_ct + a_Q sin(πt/2T_b) sin ω_ct.
• Constant envelope; main lobe 1.5R_b wide with side lobes falling as 1/f⁴; BER same as BPSK.

GMSK

Gaussian MSK passes the NRZ data through a Gaussian low-pass filter before MSK modulation.

data->[Gaussian LPF, BT]->[FM, h = 0.5]-> GMSK

• The filter smooths phase changes, reducing side lobes and adjacent channel interference.
• BT (3 dB bandwidth × bit time) sets the trade-off: GSM uses BT = 0.3. Smaller BT means narrower spectrum but more ISI.
• Keeps the constant envelope, so efficient class-C amplifiers can be used. Used in GSM, DECT.

Advantages of spread spectrum in wireless communication

1. Interference and jamming rejection – narrowband interference is spread at the receiver and reduced by the processing gain.
2. Multipath resistance – delayed copies more than one chip late are uncorrelated with the code; a RAKE receiver can even combine them for diversity gain.
3. Multiple access (CDMA) – many users share the same band at the same time using different codes; universal frequency reuse.
4. Low probability of interception – the signal is below the noise floor and looks like noise without the code.
5. Privacy/security – the PN code acts as a key.
6. Graceful degradation – capacity is soft-limited; extra users only raise the noise level slightly.
7. Accurate ranging – fast chips give good timing resolution (used in GPS).

Both are spread spectrum methods that use a PN code to spread a narrowband signal over a wide band.

Direct sequence (DSSS)

Each data bit is multiplied by a fast PN chip sequence (T_c ≪ T_b), and the product BPSK-modulates the carrier.

data->(x)->(x)-> s(t)      r(t)->(x)->(x)->[∫]-> data
       ^    ^                     ^    ^
     [PN] [cos]                 [PN] [cos]

The receiver multiplies by the same synchronized code to despread the signal. Processing gain PG = T_b/T_c. Used in IS-95 and 802.11b.

Frequency hopping (FHSS)

The carrier frequency of an FSK signal hops among many channels in a PN-controlled pattern.

data->[FSK]->(x)-> FH    FH->(x)->[FSK demod]-> data
              ^               ^
       [Synth <- PN]   [Synth <- PN]

The receiver hops in step to "de-hop" the signal. Slow or fast hopping. Used in Bluetooth.

DPSK transmitter and receiver

Differential PSK (DPSK) is a non-coherent form of BPSK. The data is carried by the change of phase between successive bits, so the receiver does not need a coherent carrier reference.

Transmitter

m_k --->[XNOR]------+-----> [BPSK ] --> DPSK
          ^         | d_k   [mod  ]
          |         |          ^
          +-[Delay T_b]<-+   cos ωct
            d_(k−1)

• Differential encoding: d_k = NOT(m_k ⊕ d_{k−1}). A "1" keeps the previous phase; a "0" changes it by π.
• The encoded sequence d_k then BPSK-modulates the carrier.

Example (reference bit d₀ = 1):

Receiver

r(t)->[BPF]-+->(x)->[∫ over T_b]->[Decision]-> m_k
            |   ^
            +->[Delay T_b]

The received signal is multiplied by itself delayed by one bit. If the two bits have the same phase, the integrator output is positive (bit 1); if opposite, it is negative (bit 0). DPSK is simpler than coherent BPSK, but needs about 3 dB more E_b/N₀ for the same BER: P_b = ½ exp(−E_b/N₀).

Pseudo-noise (PN) sequence

A PN sequence is a deterministic, periodic binary sequence that looks random. It is usually generated by an m-stage linear feedback shift register; a maximal-length (m-) sequence has period N = 2ᵐ − 1. Properties of an m-sequence:

• Balance: number of 1s exceeds number of 0s by one.
• Run property: half the runs have length 1, a quarter length 2, and so on.
• Autocorrelation: 1 at zero shift and −1/N at any other shift (noise-like).

Why it is used

• To spread and despread signals in DSSS and to set hop patterns in FHSS.
• Its sharp autocorrelation lets the receiver synchronize and reject delayed multipath and interference.
• Different (low cross-correlation) codes give each user a separate channel in CDMA, and provide privacy.

Offset QPSK (OQPSK) is QPSK in which the Q bit stream is delayed by one bit period T_b (half a symbol) relative to the I stream. Because I and Q never change at the same instant, the phase can change by at most ±90°, never 180°. This keeps the envelope nearly constant after filtering and non-linear amplification.

OQPSK transmitter

           +->[I: even bits]------------>(x)--+
           |   (2T_b each)                ^   |
bits->[S/P]                          cos ωct (+)->OQPSK
           |                              v   |
           +->[Q: odd bits]->[Delay T_b]->(x)-+
                                     sin ωct

1. NRZ data is split into I (even) and Q (odd) streams, each of duration 2T_b.
2. The Q stream is delayed by T_b.
3. I modulates cos ω_ct, Q modulates sin ω_ct, and the two are added.

Timing

I: |  a0  |  a2  |  a4  |
Q:    |  a1  |  a3  |  a5  |
      <-Tb->

OQPSK receiver

       +->(x)->[∫ 2T_b]->[Decide]-----------+
       |  cos ωct                           |
r(t)-->+                                 [P/S]-> bits
       |  sin ωct                           |
       +->(x)->[∫ 2T_b, offset T_b]->[Decide]+

Coherent I and Q correlators recover each stream; the Q integrator is offset by T_b. Decisions are combined by a parallel-to-serial converter. BER equals that of QPSK/BPSK: Q(√(2E_b/N₀)).

Why π/4-QPSK is preferred over OQPSK

In π/4-QPSK, two QPSK constellations rotated by 45° are used alternately, so each symbol causes a phase change of ±45° or ±135°.

• Non-coherent (differential) detection is possible. π/4-QPSK can be differentially encoded and detected, which avoids carrier recovery and works well in fast-fading mobile channels. OQPSK needs coherent detection.
• The maximum phase jump (135°) is smaller than QPSK's 180°, so envelope fluctuation is still low (between QPSK and OQPSK).
• There is always a phase change every symbol, which helps timing recovery.

For these reasons π/4-DQPSK was chosen for IS-54/IS-136 (US digital cellular) and PDC (Japan).

Principle of OFDM

Orthogonal Frequency Division Multiplexing splits a high-rate stream into N parallel low-rate streams, each on its own subcarrier. The subcarriers are spaced by Δf = 1/T_s, which makes them orthogonal: over one symbol period, the integral of the product of any two different subcarriers is zero.

∫₀^Ts cos(2π f_i t) cos(2π f_j t) dt = 0,  i ≠ j
f_k = f_0 + k/T_s

So the spectra may overlap, yet each subcarrier peak falls at the nulls of all others, and no inter-carrier interference occurs.

 |H|   /\  /\  /\  /\
      /  \/  \/  \/  \     overlapping sinc spectra
     /   /\  /\  /\   \    peak of one = null of others
 ---+---+---+---+---+---> f
       Δf=1/Ts

Key ideas:

• Long symbols: each subcarrier's symbol is N times longer than the original, so the delay spread is small compared with T_s and each subcarrier sees flat fading.
• IFFT/FFT: modulation and demodulation of all subcarriers is done by an N-point IFFT at the transmitter and FFT at the receiver.
• Cyclic prefix: a copy of the end of each symbol is placed at its start; if it is longer than the delay spread, ISI is removed and a one-tap equalizer per subcarrier is enough.

Types of spread spectrum modulation

1. Direct sequence SS (DSSS): data is multiplied by a fast PN chip sequence and BPSK modulated. The receiver multiplies by the same code to despread. Example: IS-95, 802.11b.
2. Frequency hopping SS (FHSS): the carrier of an FSK signal jumps among many frequencies in a PN-defined order; slow or fast hopping. Example: Bluetooth.
3. Time hopping SS (THSS): data is sent in short bursts in time slots chosen by a PN code within each frame. Example: impulse-radio UWB.
4. Hybrid systems: combinations such as DS/FH, which mix the advantages of both (e.g. military radios).

Gaussian Minimum Shift Keying (GMSK) is a constant-envelope digital modulation obtained by passing NRZ data through a Gaussian low-pass filter before MSK (FM with modulation index h = 0.5).

data->[Gaussian LPF (BT)]->[FM/VCO, h = 0.5]-> GMSK

• The filter smooths the phase path, so side lobes and adjacent-channel interference are much lower than MSK.
• BT product (3 dB filter bandwidth × bit period) sets the trade-off: smaller BT gives a narrower spectrum but more ISI. GSM uses BT = 0.3.
• Constant envelope allows efficient non-linear (class C) amplifiers.
• Can be detected coherently (like MSK) or non-coherently with an FM discriminator; GSM uses an MLSE equalizer to handle the ISI.
• Used in GSM (270.833 kbps in 200 kHz), DECT and, in GFSK form, Bluetooth.

MSK and GMSK

• MSK is continuous phase FSK with modulation index h = 0.5 (tone spacing R_b/2). Phase changes smoothly by ±π/2 per bit; constant envelope; can be seen as OQPSK with half-sine pulse shaping.
• GMSK is MSK whose data first passes a Gaussian low-pass filter (BT = 0.3 in GSM). This makes the spectrum very compact while keeping the constant envelope.

OFDM modulator

bits->[Map QAM]->[S/P]->[IFFT]->[P/S]->[Add CP]
                         N-pt               |
                                            v
                             RF <-[Up-conv]<-[DAC]

• Bits are mapped to QAM/PSK symbols and split into N parallel streams.
• The IFFT places each symbol on a separate orthogonal subcarrier (spacing 1/T_s) and produces one time-domain OFDM symbol.
• After P/S conversion a cyclic prefix longer than the channel delay spread is added, then DAC and RF up-conversion.

OFDM demodulator

RF->[Down-conv]->[ADC]->[Remove CP]->[S/P]->[FFT]
                                              |
                                              v
bits<-[Demap]<-[P/S]<-[One-tap equalizer (pilots)]

• After down-conversion and sampling, the CP is removed (it carries the ISI).
• The FFT separates the subcarriers: Y_k = H_k X_k + N_k.
• Each subcarrier is equalized by dividing by its channel estimate H_k, then demapped to bits.

OFDM turns a frequency-selective channel into many flat sub-channels, so it handles high data rates in multipath (Wi-Fi, LTE, 5G).