Digital Modulation Techniques

Quadriphase-shift keying (QPSK) is a digital modulation in which the carrier takes one of four phases, each carrying a pair of bits (a dibit). It sends 2 bits per symbol, so it needs half the bandwidth of BPSK for the same bit rate, with the same bit error probability.

Signal set

With symbol duration $T = 2T_b$ and symbol energy $E$:

$$s_i(t) = \sqrt{\frac{2E}{T}}\cos\left[2\pi f_ct + (2i-1)\frac{\pi}{4}\right], \quad 0 \le t \le T,\ i = 1, 2, 3, 4$$

Expanding the cosine:

$$s_i(t) = \sqrt{E}\cos\left[(2i-1)\frac{\pi}{4}\right]\phi_1(t) - \sqrt{E}\sin\left[(2i-1)\frac{\pi}{4}\right]\phi_2(t)$$

with two orthonormal basis functions

$$\phi_1(t) = \sqrt{\frac{2}{T}}\cos(2\pi f_ct), \qquad \phi_2(t) = \sqrt{\frac{2}{T}}\sin(2\pi f_ct), \quad 0 \le t \le T$$

($f_c$ is a multiple of $1/T$, so $\int_0^T\phi_1\phi_2\,dt = 0$ and each has unit energy.)

Coordinates $s_{i1} = \sqrt{E}\cos[(2i-1)\pi/4]$, $s_{i2} = -\sqrt{E}\sin[(2i-1)\pi/4]$:

Signal space diagram

              phi2
               |
   01  *       |       *  11
  (-a,+a)      |      (+a,+a)
               |
 --------------+--------------- phi1
               |
   00  *       |       *  10
  (-a,-a)      |      (+a,-a)
               |
        a = sqrt(E/2)

The four points lie on a circle of radius $\sqrt{E}$. Gray coding makes adjacent points differ by one bit. The distance between adjacent points is $d = \sqrt{2E} = 2\sqrt{E_b}$ (with $E = 2E_b$), the same as BPSK, so the bit error probability is the same:

$$P_e(\text{bit}) = \frac12\,\text{erfc}\sqrt{\frac{E_b}{N_0}}$$

QPSK modulator

          +-> a1(t) -->(x)----+
binary    |   (odd bits)   ^  |
data --[serial to]         |  |
 b(t)   [parallel ]  phi1(t)  (+)--> QPSK s(t)
          |   (even bits)  |  |
          +-> a2(t) -->(x)----+
                           ^
                        phi2(t)

1. The input bit stream $b(t)$ is converted to polar NRZ form ($1 \to +\sqrt{E_b}$, $0 \to -\sqrt{E_b}$).
2. A demultiplexer (serial-to-parallel converter) splits it into odd bits $a_1(t)$ and even bits $a_2(t)$, each of duration $T = 2T_b$.
3. $a_1(t)$ multiplies the in-phase carrier $\phi_1(t)$ and $a_2(t)$ multiplies the quadrature carrier $\phi_2(t)$, giving two BPSK signals.
4. The sum of the two BPSK signals is the QPSK signal:

$$s(t) = a_1(t)\phi_1(t) + a_2(t)\phi_2(t)$$

Since each branch runs at $R_b/2$, the null-to-null bandwidth is $B = 2/T = R_b$ (BPSK needs $2R_b$).

QPSK demodulator (coherent receiver)

          +->(x)->[integrate 0..T]->[decide]->a1^-+
  x(t)    |   ^                     thresh 0      |
 --------+  phi1(t)                         [parallel]-> b^
          |                                 [to serial]
          +->(x)->[integrate 0..T]->[decide]->a2^-+
              ^                     thresh 0
            phi2(t)

The received signal is $x(t) = s_i(t) + w(t)$, with $w(t)$ white Gaussian noise of PSD $N_0/2$.

1. Two correlators multiply $x(t)$ by locally generated coherent references $\phi_1(t)$ and $\phi_2(t)$ and integrate over one symbol:

$$x_1 = \int_0^T x(t)\phi_1(t)\,dt = s_{i1} + w_1, \qquad x_2 = \int_0^T x(t)\phi_2(t)\,dt = s_{i2} + w_2$$

Because of orthogonality, the in-phase branch sees only $s_{i1}$ and the quadrature branch only $s_{i2}$; $w_1, w_2$ are independent Gaussian with variance $N_0/2$. 2. Each output is compared with a threshold of zero: $x_1 > 0 \Rightarrow$ 1, else 0 (same for $x_2$). This is the minimum-distance decision in the signal space; the decision regions are the four quadrants. 3. A multiplexer (parallel-to-serial converter) combines the two decided bit streams into the original sequence.

Error performance

Each branch is a BPSK detector with $E_b = E/2$:

$$P_e(\text{bit}) = \frac12\,\text{erfc}\sqrt{\frac{E_b}{N_0}}, \qquad P_e(\text{symbol}) \approx \text{erfc}\sqrt{\frac{E}{2N_0}}$$

So QPSK gives the error performance of BPSK with twice the bandwidth efficiency (2 bits/s/Hz in theory).

Binary FSK (BFSK) sends a bit by switching the carrier between two frequencies: one frequency for symbol 1 and another for symbol 0, while the amplitude stays constant.

$$s_i(t)=\begin{cases}\sqrt{\dfrac{2E_b}{T_b}}\cos(2\pi f_i t), & 0\le t\le T_b\\[4pt] 0, & \text{elsewhere}\end{cases}\qquad i=1,2$$

with $f_i=\dfrac{n_c+i}{T_b}$ ($n_c$ an integer), so that the two tones are orthogonal over one bit. Symbol 1 is sent by $s_1(t)$ (frequency $f_1$) and symbol 0 by $s_2(t)$ (frequency $f_2$).

Modulator

The binary data in on-off form is applied to one product modulator, and its inverse to the other. Exactly one oscillator output reaches the adder in each bit interval.

 b(t) on-off
   |---------------->(X)<---- cos 2*pi*f1*t
   |                  |
   |                  v
   |                 (+)------> BFSK s(t)
   |                  ^
   v                  |
 [Inverter]-------->(X)<---- cos 2*pi*f2*t

• Bit 1: upper channel ON, lower OFF, output is tone $f_1$.
• Bit 0: inverter turns lower channel ON, output is tone $f_2$.
• If the two oscillators are phase-synchronised (or one VCO is used), the phase is continuous (CPFSK).

Coherent demodulator

The receiver uses two correlators, one matched to each basis function, and compares their outputs.

         +->(X)->[int 0..Tb]--x1--+
         |   ^ phi1(t)            |+
 x(t) ---+                       (S)--> l = x1 - x2
         |   v phi2(t)            |-      |
         +->(X)->[int 0..Tb]--x2--+       v
                              [l > 0 -> 1, l < 0 -> 0]

$\phi_1(t)=\sqrt{2/T_b}\cos 2\pi f_1t$ and $\phi_2(t)=\sqrt{2/T_b}\cos 2\pi f_2t$. If $l=x_1-x_2>0$ the receiver decides 1, otherwise 0. A non-coherent version uses band-pass filters at $f_1$ and $f_2$ followed by envelope detectors and a comparator.

Signal space diagram

BFSK is a two-dimensional scheme ($N=2$, $M=2$). The signal vectors are

$$\mathbf{s}_1=\begin{bmatrix}\sqrt{E_b}\\0\end{bmatrix},\qquad \mathbf{s}_2=\begin{bmatrix}0\\\sqrt{E_b}\end{bmatrix}$$

   phi2
    |
 s2 *  (0, sqrt Eb)       region Z2
    |   \
    |     \  decision boundary
    |       \  (x1 = x2 line)
    |         \
    +-----------*---------> phi1
  0          s1 (sqrt Eb, 0)
                 region Z1

• Distance between points: $d_{12}=\sqrt{2E_b}$ (smaller than $2\sqrt{E_b}$ for BPSK).
• Decision boundary is the line $x_1=x_2$ at 45 degrees.
• Bit error probability (coherent): $P_e=\tfrac12\,\text{erfc}\sqrt{E_b/2N_0}$, so BFSK needs 3 dB more $E_b/N_0$ than BPSK for the same $P_e$.
• Bandwidth is about $\lvert f_1-f_2\rvert+2/T_b$, larger than BPSK.

Minimum Shift Keying (MSK) is continuous-phase binary FSK in which the two tones are separated by the minimum spacing that still keeps them orthogonal, $\Delta f=f_1-f_2=\dfrac{1}{2T_b}$ (deviation ratio $h=0.5$).

Signal

$$s(t)=\sqrt{\frac{2E_b}{T_b}}\cos\left[2\pi f_c t+\theta(t)\right],\qquad \theta(t)=\theta(0)\pm\frac{\pi}{2T_b}t$$

• Bit 1: phase increases by $+\pi/2$ over the bit ($f_1=f_c+\tfrac{1}{4T_b}$).
• Bit 0: phase decreases by $-\pi/2$ over the bit ($f_2=f_c-\tfrac{1}{4T_b}$).
• The phase is continuous at bit boundaries, so there are no sudden jumps.

Expanding in quadrature form, MSK equals offset QPSK with half-sinusoid pulse shaping:

$$s(t)=\sqrt{\tfrac{2E_b}{T_b}}\cos\theta(t)\cos 2\pi f_ct-\sqrt{\tfrac{2E_b}{T_b}}\sin\theta(t)\sin 2\pi f_ct$$

Modulator

 data -> [S/P: odd/even bits, each 2Tb long]
            |  I bits             | Q bits (offset Tb)
            v                     v
     (X) cos(pi t/2Tb)     (X) sin(pi t/2Tb)
            |                     |
     (X) cos 2pi fc t      (X) sin 2pi fc t
            |                     |
            +-------->(+)<--------+
                       |
                     MSK out

Example

Data 1 1 0 1, starting phase $\theta(0)=0$:

The phase moves along straight lines (phase trellis) and never jumps. For $T_b=1\ \mu\text{s}$ the tones are $f_c\pm 250$ kHz, a spacing of 500 kHz $=1/(2T_b)$.

Features

• Constant envelope, so efficient non-linear amplifiers can be used.
• Main lobe is 1.5 times wider than QPSK, but sidelobes are much lower, falling as $1/f^4$ (less adjacent-channel interference).
• Same $P_e$ as BPSK/QPSK with coherent detection: $\tfrac12\text{erfc}\sqrt{E_b/N_0}$.
• Gaussian filtering of the data before MSK gives GMSK (used in GSM).

Quadrature Amplitude Modulation (QAM) is an M-ary scheme in which both the amplitude and the phase of the carrier change. Two independent amplitude-modulated carriers, $\cos 2\pi f_ct$ (in-phase, I) and $\sin 2\pi f_ct$ (quadrature, Q), are added so that each symbol carries $\log_2 M$ bits.

$$s_k(t)=\sqrt{\frac{2E_0}{T}}\,a_k\cos 2\pi f_ct-\sqrt{\frac{2E_0}{T}}\,b_k\sin 2\pi f_ct,\qquad 0\le t\le T$$

where $(a_k,b_k)$ are integer levels, e.g. $\pm1,\pm3$ for 16-QAM, and $E_0$ is the energy of the lowest-amplitude signal.

Modulator

          +-> [2-to-L level] -> a_k -->(X)<- cos wc t
 binary   |                               |
 data -->[S/P]                           (+)--> QAM
          |                               |
          +-> [2-to-L level] -> b_k -->(X)<- -sin wc t

1. The serial data is split into two streams (half the bits each).
2. Each stream is grouped into $\log_2\sqrt{M}$ bits and converted to one of $L=\sqrt{M}$ levels.
3. The levels multiply the I and Q carriers and the products are added.

Demodulator (coherent)

          +->(X)->[int 0..T]->[L-level decision]-+
          |   ^ cos wc t                         |
 r(t) ----+                                    [P/S]-> data
          |   v sin wc t                         |
          +->(X)->[int 0..T]->[L-level decision]-+

1. Two correlators (multiply and integrate) recover the I and Q amplitudes; orthogonality of cos and sin removes cross-talk.
2. Each output is compared with $L-1$ thresholds to decide the level.
3. Level-to-bits conversion and a parallel-to-serial converter give the data back.

A carrier recovery circuit is needed because detection is coherent.

16-QAM constellation

            Q
   *    *   |   *    *     +3
            |
   *    *   |   *    *     +1
 -----------+-----------> I
   *    *   |   *    *     -1
            |
   *    *   |   *    *     -3
  -3   -1      +1   +3

16 points, 4 bits per symbol; with Gray coding, neighbours differ by one bit. QAM gives higher bandwidth efficiency than M-ary PSK for the same $M$ because its points are spread more evenly over the plane.

Significance of multilevel modulation

In multilevel (M-ary) modulation each symbol carries $n=\log_2 M$ bits, instead of one bit as in binary schemes.

• Bandwidth saving: symbol rate is $R_b/\log_2 M$, so bandwidth falls by the factor $\log_2 M$ (QPSK needs half the bandwidth of BPSK).
• Higher data rate on a band-limited channel (telephone line, radio channel) for the same bandwidth.
• Higher spectral efficiency (bits/s/Hz); used in modems, Wi-Fi, LTE (16/64-QAM).
• Cost: for a fixed power, points get closer as $M$ rises, so error probability increases or more power is needed; the receiver is more complex.

QPSK

Quadrature PSK sends two bits (a dibit) per symbol using four carrier phases:

$$s_i(t)=\sqrt{\frac{2E}{T}}\cos\left[2\pi f_ct+(2i-1)\frac{\pi}{4}\right],\quad i=1,2,3,4,\ 0\le t\le T$$

with $T=2T_b$ and $E=2E_b$. Expanding, $s_i(t)=\sqrt{E}\cos[(2i-1)\tfrac{\pi}{4}]\phi_1(t)-\sqrt{E}\sin[(2i-1)\tfrac{\pi}{4}]\phi_2(t)$, so QPSK is two BPSK signals on quadrature carriers.

Transmitter

 bits -> [NRZ level encoder: 1->+1, 0->-1]
               |
            [Demux S/P]
           /           \
     a1(t) odd      a2(t) even
        |               |
       (X)<-phi1(t)    (X)<-phi2(t)
        |               |
        +----->(+)<-----+
                |
              QPSK

$\phi_1(t)=\sqrt{2/T}\cos 2\pi f_ct$, $\phi_2(t)=\sqrt{2/T}\sin 2\pi f_ct$. Each branch is a BPSK signal at half the bit rate.

Receiver

         +->(X)->[int 0..T]->[x1>0 ? 1:0]-+
         |   ^ phi1                       |
 x(t) ---+                              [Mux P/S]-> bits
         |   v phi2                       |
         +->(X)->[int 0..T]->[x2>0 ? 1:0]-+

Each correlator output is compared with a zero threshold; the two decisions are multiplexed back into one stream.

• Bandwidth: half that of BPSK for the same bit rate.
• Bit error probability: $P_e=\tfrac12\text{erfc}\sqrt{E_b/N_0}$, the same as BPSK.

Binary Phase Shift Keying (BPSK) sends a bit by shifting the carrier phase by $180^\circ$: phase $0$ for symbol 1 and phase $\pi$ for symbol 0. Amplitude and frequency stay constant.

$$s_1(t)=\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_ct,\qquad s_2(t)=-\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_ct,\qquad 0\le t\le T_b$$

$f_c=n_c/T_b$ so each bit holds a whole number of cycles. $s_1$ and $s_2$ are antipodal signals.

Waveform (data 1 0 1)

 data:   1        0        1
       /\  /\   \  /\  /   /\  /\
      /  \/  \   \/  \/   /  \/  \
              phase flips at 1->0 and 0->1

Modulator

 binary --> [NRZ polar encoder] --> b(t) = +-1
                                     |
                        (X) <-- sqrt(2/Tb) cos 2pi fc t
                         |
                       BPSK s(t)

The polar NRZ signal (+1 for 1, -1 for 0) multiplies the carrier, so the product modulator flips the phase whenever the data sign changes.

Coherent demodulator

 x(t) -->(X)--> [integrate 0..Tb] --> x1 --> [decision]
          ^                                  x1>0 -> 1
     phi1(t) from carrier recovery           x1<0 -> 0

The correlator gives $x_1=\int_0^{T_b}x(t)\phi_1(t)dt$, which equals $\pm\sqrt{E_b}$ plus noise.

Signal space diagram

One basis function is enough ($N=1$):

$$\phi_1(t)=\sqrt{\frac{2}{T_b}}\cos 2\pi f_ct,\qquad s_1(t)=\sqrt{E_b}\,\phi_1(t),\quad s_2(t)=-\sqrt{E_b}\,\phi_1(t)$$

   region Z2        |        region Z1
      s2            |            s1
 -----*-------------+-------------*-----> phi1
   -sqrt(Eb)        0         +sqrt(Eb)
                 boundary

• Distance between points: $d=2\sqrt{E_b}$, the largest possible for a given energy.
• Decision threshold at zero.
• $P_e=\tfrac12\,\text{erfc}\sqrt{E_b/N_0}$, the best among binary schemes.
• Bandwidth $\approx 2/T_b=2R_b$ (null to null).

Quadrature Amplitude Modulation (QAM) sends two independent message signals on the same carrier frequency, using two carriers of the same frequency that are $90^\circ$ apart (cos and sin). It is also called quadrature-carrier multiplexing. In digital form (M-ary QAM) both amplitude and phase change from symbol to symbol.

$$s(t)=A_c\,m_1(t)\cos 2\pi f_ct+A_c\,m_2(t)\sin 2\pi f_ct$$

Why QAM is necessary

• Bandwidth saving: two DSB-SC signals occupy the bandwidth of one, i.e. $2W$ for two messages. Its spectral efficiency equals SSB without needing sharp sideband filters.
• Higher data rate: in digital form, M-ary QAM carries $\log_2 M$ bits per symbol (16-QAM: 4 bits), so more data fits a band-limited channel.
• Better noise performance than M-ary PSK for the same $M$, because points are spread over the whole plane, not only a circle.
• Used in colour TV (chrominance), telephone modems, Wi-Fi, cable TV, LTE.

Generation

 m1(t) ---------->(X)<--- Ac cos wc t
                   |
                  (+)-------> s(t) QAM
                   |
 m2(t) ---------->(X)<--- Ac sin wc t
                        (-90 deg shift of
                         the same oscillator)

Two product modulators share one oscillator; one gets the carrier directly and the other through a $-90^\circ$ phase shifter. Their outputs are added. In digital QAM, $m_1$ and $m_2$ are multi-level signals $a_k$, $b_k$ produced by splitting the bit stream (serial-to-parallel) and converting groups of bits to levels.

Detection (coherent)

        +->(X)->[LPF]-> (Ac/2) m1(t)
        |   ^ cos wc t
 s(t) --+
        |   v sin wc t
        +->(X)->[LPF]-> (Ac/2) m2(t)

In-phase branch:

$$\begin{aligned} s(t)\cos\omega_ct &= A_cm_1\cos^2\omega_ct+A_cm_2\sin\omega_ct\cos\omega_ct\\ &=\frac{A_c}{2}m_1(t)+\frac{A_c}{2}m_1\cos 2\omega_ct+\frac{A_c}{2}m_2\sin 2\omega_ct \end{aligned}$$

The LPF removes the $2\omega_c$ terms, leaving $\tfrac{A_c}{2}m_1(t)$. Similarly the quadrature branch gives $\tfrac{A_c}{2}m_2(t)$.

The local carrier must have exactly the right phase and frequency. A phase error $\phi$ gives $\tfrac{A_c}{2}[m_1\cos\phi - m_2\sin\phi]$, i.e. cross-talk between channels. So a pilot or carrier recovery circuit (e.g. Costas loop) is used. In digital QAM the LPFs are replaced by integrate-and-dump correlators followed by multi-level decision devices.

Differential PSK (DPSK) is the non-coherent form of BPSK. The information is carried by the phase change between consecutive bits, so the receiver uses the previous received bit as its phase reference and needs no locally generated coherent carrier.

DPSK signal (recap)

The data $b_k$ is first differentially encoded:

$$d_k=\overline{b_k\oplus d_{k-1}}$$

i.e. if $b_k=1$ the encoded bit is unchanged, if $b_k=0$ it toggles. Then $d_k$ drives a BPSK modulator (1: phase 0, 0: phase $\pi$).

Demodulation method: delay-and-multiply (correlation) receiver

 r(t)->[BPF]-+------->(X)->[int 0..Tb]->[decide]-> b_k
             |         ^                 y>0 -> 1
             |         |                 y<0 -> 0
             +->[delay Tb]

Working:

1. The received signal in bit $k$ is $r_k(t)=A\cos(\omega_ct+\theta_k)$ and in the previous bit, after delay $T_b$, $r_{k-1}(t)=A\cos(\omega_ct+\theta_{k-1})$.
2. The multiplier gives

$$r_k\,r_{k-1}=\frac{A^2}{2}\cos(\theta_k-\theta_{k-1})+\frac{A^2}{2}\cos(2\omega_ct+\theta_k+\theta_{k-1})$$

3. The integrator (low-pass action) removes the $2\omega_c$ term and keeps $y=\tfrac{A^2T_b}{2}\cos(\theta_k-\theta_{k-1})$.
4. If the phase did not change, $y>0$, decide $b_k=1$; if it changed by $\pi$, $y<0$, decide $b_k=0$.

Any unknown fixed carrier phase appears in both $\theta_k$ and $\theta_{k-1}$ and cancels, which is why no carrier recovery is needed.

Example

The output equals the original data.

Remarks

• Error probability: $P_e=\tfrac12 e^{-E_b/N_0}$, about 1 dB worse than coherent BPSK at $P_e=10^{-4}$.
• Errors tend to occur in pairs, because a wrong bit is also the reference for the next bit.
• The other method is coherent detection of $d_k$ (as in BPSK) followed by a differential decoder ($b_k=\overline{d_k\oplus d_{k-1}}$); it gives slightly better $P_e$ but needs carrier recovery.

ASK modulator

In Amplitude Shift Keying (on-off keying) the carrier is sent for bit 1 and switched off for bit 0: $s(t)=b(t)\,A\cos 2\pi f_ct$, with $b(t)\in\{1,0\}$.

 binary (unipolar NRZ) b(t) ---->(X)----> ASK
                                  ^
                     A cos 2 pi fc t (oscillator)

The product modulator (or simply a switch) passes the carrier only during 1s.

ASK demodulators

Coherent:

 ASK -->(X)-->[integrate 0..Tb]-->[compare with Vth]--> data
         ^
      cos wc t (carrier recovery)

Non-coherent (envelope detection):

 ASK ->[BPF at fc]->[envelope det]->[comparator]-> data

The output is high when the carrier is present (1) and near zero when it is absent (0); the threshold is about half the peak.

FSK modulator

In Frequency Shift Keying bit 1 is sent at $f_1$ and bit 0 at $f_2$.

 b(t) --------------->(X)<-- cos w1 t --+
   |                                    (+)--> FSK
   +-->[Inverter]---->(X)<-- cos w2 t --+

Equivalently, the polar NRZ data can drive a VCO, whose output frequency switches between $f_1$ and $f_2$ (giving continuous phase).

FSK demodulators

Coherent:

        +->(X)<-cos w1 t ->[int 0..Tb]--+
 FSK ---+                              (-)--> l --> l>0: 1
        +->(X)<-cos w2 t ->[int 0..Tb]--+           l<0: 0

Non-coherent:

        +->[BPF f1]->[envelope det]--+
 FSK ---+                           [compare]--> data
        +->[BPF f2]->[envelope det]--+

The branch with the larger envelope decides the bit. A PLL demodulator can also be used.

Can PSK be detected with an envelope detector?

No. In binary PSK, $s(t)=\pm A\cos 2\pi f_ct$. The amplitude is $A$ for both 1 and 0; only the phase differs by $180^\circ$:

$$\text{envelope of } +A\cos\omega_ct = \text{envelope of } -A\cos\omega_ct = A$$

An envelope detector responds only to amplitude, so its output is a constant $A$ for every bit and all information is lost. PSK must be detected coherently (multiply by a locally recovered carrier and integrate, the sign gives the bit), or in differential form (DPSK) by comparing each bit with the previous one.

Why shift keying is used

Shift keying (digital carrier modulation) means switching the amplitude, frequency or phase of a high-frequency sinusoidal carrier according to the binary data. It is used because:

• Bandpass channels: radio, satellite, microwave and telephone channels pass only a band of frequencies around some centre frequency; baseband pulses (with energy near DC) cannot go through them directly.
• Practical antenna size: antennas must be about $\lambda/4$ long; a high carrier frequency makes them small.
• Multiplexing: different users can be given different carrier frequencies (FDM).
• Noise and interference control: FSK and PSK keep a constant envelope and so resist amplitude noise, and the scheme can be chosen to trade bandwidth for power.
• Long-distance transmission over wireless and existing analog lines (modems).

ASK vs FSK vs PSK

 data :   1      0      1
 ASK  : /\/\/\ ______ /\/\/\
 FSK  : /\/\/\ /\  /\ /\/\/\   (fewer cycles for 0)
 PSK  : /\/\/\ \/\/\/ /\/\/\   (phase flips)

($E_b$ is the average energy per bit; for ASK the "1" bit has energy $2E_b$.)

Quadrature Amplitude Modulation (QAM) is a digital modulation in which each symbol is sent by changing both the amplitude and phase of the carrier. It is built from two amplitude-modulated carriers of the same frequency in quadrature, so an $M$-point QAM carries $\log_2 M$ bits per symbol.

$$s_k(t)=\sqrt{\frac{2E_0}{T}}\,a_k\cos 2\pi f_ct-\sqrt{\frac{2E_0}{T}}\,b_k\sin 2\pi f_ct,\quad 0\le t\le T$$

For 16-QAM, $a_k,b_k\in\{-3,-1,+1,+3\}$.

Modulator

              +->[2-bit to 4-level]-a_k->(X)<-phi1
 bits -->[S/P]                             |
  4 bits/sym  |                           (+)--> s(t)
              +->[2-bit to 4-level]-b_k->(X)<-phi2

$\phi_1(t)=\sqrt{2/T}\cos 2\pi f_ct$, $\phi_2(t)=-\sqrt{2/T}\sin 2\pi f_ct$.

1. Serial-to-parallel converter sends 2 bits to the I branch and 2 bits to the Q branch.
2. Each 2-bit group is mapped (Gray code) to a level: 00 to -3, 01 to -1, 11 to +1, 10 to +3.
3. The levels amplitude-modulate the quadrature carriers and the outputs are summed.

Demodulator

          +->(X)->[int 0..T]->[4-level slicer]--+
          |   ^ phi1                            |
 r(t) ----+                                   [P/S]-> bits
          |   v phi2                            |
          +->(X)->[int 0..T]->[4-level slicer]--+

Each correlator recovers one coordinate ($\approx a_k\sqrt{E_0}$ and $b_k\sqrt{E_0}$ plus noise). The slicer compares it with thresholds $0,\pm2\sqrt{E_0}$, picks the nearest level, and the level-to-bits converter plus P/S give the data. Carrier and symbol timing recovery are needed (coherent detection).

16-QAM constellation (Gray coded)

               Q (phi2)
  0010  0110   |   1110  1010   +3
               |
  0011  0111   |   1111  1011   +1
 --------------+--------------> I (phi1)
  0001  0101   |   1101  1001   -1
               |
  0000  0100   |   1100  1000   -3
   -3    -1        +1    +3

First two bits select the I level, last two the Q level; neighbouring points differ by one bit, so most symbol errors cause only one bit error. The minimum distance between points is $2\sqrt{E_0}$.

Merits

• 4 bits/symbol: bandwidth is 1/4 of BPSK for the same bit rate.
• Better error performance than 16-PSK at the same average power.
• Not constant envelope, so linear amplifiers are required.

Generation of PSK

In binary Phase Shift Keying the carrier phase is $0$ for bit 1 and $\pi$ for bit 0:

$$s(t)=\pm\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_ct,\quad 0\le t\le T_b$$

 data -->[polar NRZ encoder]--> b(t)=+-1 -->(X)--> BPSK
                                            ^
                            sqrt(2/Tb) cos 2pi fc t

The polar NRZ data multiplies the carrier in a product (balanced) modulator. When $b(t)$ changes sign the output phase jumps by $180^\circ$.

 data:   1       0       1
 PSK : /\/\/\  \/\/\/  /\/\/\

Detection of PSK (coherent)

 x(t)-->(X)-->[integrate 0..Tb]--x-->[decision x>0 ? 1 : 0]
         ^
   carrier recovery (squaring loop / Costas loop)

1. The receiver regenerates a carrier exactly in phase with the transmitted one (e.g. square the signal, extract $2f_c$ with a PLL, divide by 2).
2. The product $x(t)\cos\omega_ct$ is integrated over the bit, giving $x=\pm\sqrt{E_b}$ + noise.
3. A zero-threshold decision device outputs 1 for $x>0$, 0 for $x<0$.

Signal space: two antipodal points $\pm\sqrt{E_b}$ on $\phi_1$; $P_e=\tfrac12\text{erfc}\sqrt{E_b/N_0}$. Envelope detection is impossible because the envelope is constant.

GMSK

Gaussian Minimum Shift Keying is MSK in which the NRZ data is first passed through a Gaussian low-pass filter before it frequency-modulates the carrier with $h=0.5$.

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

• The filter smooths the sharp transitions, so the phase changes gradually and the spectrum becomes very compact.
• The product $BT$ (filter 3-dB bandwidth x bit period) sets the trade-off; GSM uses $BT=0.3$.
• Some inter-symbol interference is introduced, but it is small for $BT\ge0.3$.

Advantages: narrow main lobe and very low sidelobes (less adjacent-channel interference), constant envelope so efficient class-C power amplifiers can be used (long battery life), and simple non-coherent or coherent detection. These are why GSM and DECT use GMSK.

Binary FSK transmits symbol 1 as a tone of frequency $f_1$ and symbol 0 as a tone of frequency $f_2$, each lasting $T_b$ and having constant amplitude:

$$s_i(t)=\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_it,\qquad f_i=\frac{n_c+i}{T_b},\ i=1,2,\ 0\le t\le T_b$$

The choice $f_i=(n_c+i)/T_b$ makes the tones orthogonal over a bit.

Generation

 data b(t) (on-off) ------->(X)<-- sqrt(2Eb/Tb) cos w1 t
     |                       |
     |                      (+)--------> BFSK
     |                       |
     +-->[Inverter]-------->(X)<-- sqrt(2Eb/Tb) cos w2 t

1. The binary data is in on-off (unipolar) form: 1 is "on", 0 is "off".
2. For bit 1 the upper product modulator passes the $f_1$ oscillator; the inverter blocks the lower one.
3. For bit 0 the inverter output is 1, so the $f_2$ oscillator passes.
4. The adder combines both branches; only one tone appears in each bit.

If the two oscillators are synchronised, phase is continuous at bit boundaries (CPFSK). A single VCO driven by polar data gives the same result.

Non-coherent detection

The phase of the received tone is not needed. Two filter-envelope branches compare energy at $f_1$ and $f_2$.

        +->[BPF f1]->[Env. det]--l1--+
        |                            |
 x(t) --+                      [Comparator]--> data
        |                            |   l1>l2: 1
        +->[BPF f2]->[Env. det]--l2--+   l1<l2: 0

1. Each band-pass filter (or filter matched to $\cos 2\pi f_it$) passes only its own tone.
2. The envelope detector outputs the amplitude of that tone at the end of the bit (sampled at $t=T_b$).
3. The comparator decides 1 if $l_1>l_2$, else 0.

An equivalent form uses quadrature correlators (cos and sin) in each branch and squares and adds their outputs. Error probability: $P_e=\tfrac12 e^{-E_b/2N_0}$, slightly worse than coherent BFSK but with a much simpler receiver (no carrier recovery).

Signal space diagram

Orthonormal basis: $\phi_i(t)=\sqrt{2/T_b}\cos 2\pi f_it$, $i=1,2$. Then

$$\mathbf{s}_1=(\sqrt{E_b},\,0),\qquad \mathbf{s}_2=(0,\,\sqrt{E_b})$$

  phi2
   |
   * s2 (0, sqrt Eb)     / boundary x1 = x2
   |    Z2           /
   |             /
   |         /      Z1
   |     /
   +---------------*--------> phi1
  0             s1 (sqrt Eb, 0)

• Two-dimensional, two orthogonal message points.
• Euclidean distance $d=\sqrt{2E_b}$.
• Decision boundary: the bisector $x_1=x_2$.
• Coherent $P_e=\tfrac12\text{erfc}\sqrt{E_b/2N_0}$, i.e. 3 dB worse than BPSK because of the smaller distance.

Quadrature Phase Shift Keying (QPSK) is a 4-level PSK in which each symbol carries a pair of bits (dibit) and the carrier takes one of four phases spaced $90^\circ$ apart.

$$s_i(t)=\sqrt{\frac{2E}{T}}\cos\left[2\pi f_ct+(2i-1)\frac{\pi}{4}\right],\quad i=1,\dots,4,\ 0\le t\le T$$

$T=2T_b$ is the symbol duration and $E=2E_b$ the symbol energy. Using $\cos(A+B)$:

$$s_i(t)=\sqrt{E}\cos\left[(2i-1)\tfrac{\pi}{4}\right]\phi_1(t)-\sqrt{E}\sin\left[(2i-1)\tfrac{\pi}{4}\right]\phi_2(t)$$

with $\phi_1(t)=\sqrt{2/T}\cos 2\pi f_ct$ and $\phi_2(t)=\sqrt{2/T}\sin 2\pi f_ct$. So QPSK is the sum of two BPSK signals on quadrature carriers.

Signal space (Gray coded)

            phi2
   01 *      |      * 11
              |
  ------------+------------ phi1
              |
   00 *      |      * 10
 points at (+-sqrt(E/2), +-sqrt(E/2))

QPSK modulator

 bits-->[polar NRZ encoder]-->[Demux S/P]
                               |        |
                         a1(t) odd  a2(t) even
                         (rate Rb/2) (rate Rb/2)
                               |        |
                     phi1 ->(X)      (X)<- phi2
                               |        |
                               +->(+)<--+
                                   |
                                 QPSK s(t)

1. Data is converted to polar NRZ (+1, -1).
2. The demultiplexer splits it into odd and even bits, each lasting $T=2T_b$.
3. Odd bits modulate the in-phase carrier, even bits the quadrature carrier (two BPSK signals).
4. The adder gives one of four phases; e.g. dibit 11 gives the point $(+\sqrt{E/2},+\sqrt{E/2})$, phase $7\pi/4$.

QPSK demodulator (coherent)

          +->(X)->[int 0..T]-x1->[x1>0?1:0]--+
          |   ^ phi1                          |
 x(t) ----+                                 [Mux P/S]-> bits
          |   v phi2                          |
          +->(X)->[int 0..T]-x2->[x2>0?1:0]--+

1. The received signal goes to two correlators fed with the recovered carriers $\phi_1$ and $\phi_2$.
2. Because $\phi_1\perp\phi_2$, each correlator sees only its own BPSK component: $x_1=\pm\sqrt{E/2}$, $x_2=\pm\sqrt{E/2}$ (plus noise).
3. Each output is compared with a zero threshold to give the odd and even bit.
4. The multiplexer recombines them into the original serial stream.

Properties

QPSK doubles bandwidth efficiency without loss in bit-error performance, which is why it is used in satellite links, DVB-S and CDMA systems. Offset QPSK (Q branch delayed by $T_b$) avoids $180^\circ$ jumps.

Design goals of digital modulation

A good digital modulation scheme aims to:

1. Maximise data rate for the available channel.
2. Minimise probability of symbol/bit error for a given $E_b/N_0$.
3. Minimise transmitted power (important for battery and satellite systems).
4. Minimise channel bandwidth (high spectral efficiency, bits/s/Hz).
5. Maximise resistance to interference and fading.
6. Minimise circuit complexity and cost.

These goals conflict (e.g. higher-level schemes save bandwidth but need more power), so the choice is a trade-off.

Coherent binary PSK

In BPSK the carrier phase is shifted by $180^\circ$ between the two symbols:

$$s_1(t)=\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_ct\ \ (\text{symbol }1),\qquad s_2(t)=-\sqrt{\frac{2E_b}{T_b}}\cos 2\pi f_ct\ \ (\text{symbol }0)$$

for $0\le t\le T_b$, where $E_b$ is the bit energy and $f_c=n_c/T_b$. The two signals are antipodal.

Signal space diagram

Only one orthonormal basis function is required:

$$\phi_1(t)=\sqrt{\frac{2}{T_b}}\cos 2\pi f_ct,\quad 0\le t\le T_b$$

so $s_1(t)=\sqrt{E_b}\,\phi_1(t)$ and $s_2(t)=-\sqrt{E_b}\,\phi_1(t)$. The coordinates are

$$s_{11}=\int_0^{T_b}s_1(t)\phi_1(t)dt=+\sqrt{E_b},\qquad s_{21}=-\sqrt{E_b}$$

        Z2 (decide 0)  |  Z1 (decide 1)
   s2                   |                   s1
 ---*-------------------+-------------------*---> phi1
 -sqrt(Eb)              0              +sqrt(Eb)
                 decision boundary

Distance $d_{12}=2\sqrt{E_b}$. The decision boundary is the midpoint, $x_1=0$.

Modulator

 binary data -->[Polar NRZ level encoder]--> b(t)
                 1 -> +sqrt(Eb), 0 -> -sqrt(Eb)
                                    |
                                   (X)<-- phi1(t)
                                    |
                                 BPSK s(t)

The level encoder converts 1 and 0 to $+\sqrt{E_b}$ and $-\sqrt{E_b}$; the product modulator multiplies by $\phi_1(t)$. The result is a constant-envelope carrier whose phase flips with the data.

Demodulator (coherent detector)

 x(t) -->(X)-->[ integral 0..Tb ]--x1--> [Decision device]
          ^                              x1 > 0 -> 1
        phi1(t)                          x1 < 0 -> 0
   (from carrier and bit-timing recovery)

1. The received signal $x(t)=s_i(t)+w(t)$ is multiplied by a locally generated $\phi_1(t)$, exactly in phase with the transmitted carrier.
2. The integrator over one bit gives $x_1=\pm\sqrt{E_b}+n$, where $n$ is Gaussian with variance $N_0/2$.
3. The decision device compares $x_1$ with zero.

Performance

$$P_e=\frac12\,\text{erfc}\left(\sqrt{\frac{E_b}{N_0}}\right)$$

BPSK has the lowest error probability of all binary schemes (3 dB better than coherent BFSK). Its null-to-null bandwidth is $2/T_b$. Its drawback is the need for an accurate coherent reference (Costas or squaring loop).

Why DPSK is preferred over PSK

Coherent PSK needs a local carrier exactly in phase with the received carrier; carrier recovery circuits are complex and suffer from a $180^\circ$ phase ambiguity. DPSK sends information in the phase difference between successive bits, so the previous bit serves as the reference. No coherent carrier is needed, the receiver is simpler, and phase ambiguity does not matter. The cost is about 1 dB more $E_b/N_0$ ($P_e=\tfrac12e^{-E_b/N_0}$).

Modulator

 b_k -->[XNOR]---d_k--->[level shift]-->(X)--> DPSK
          ^       |      1->+1,0->-1     ^
          |   [delay Tb]           A cos wc t
          +--d_(k-1)--+

Differential encoding: $d_k=\overline{b_k\oplus d_{k-1}}$ (bit 1: no change, bit 0: change), starting from an arbitrary reference bit. The encoded sequence then drives a BPSK modulator.

Example, reference $d_0=1$:

Demodulator

 r(t)-->[BPF]--+-------------->(X)-->[int 0..Tb]-->[y>0?1:0]
               |                ^
               +-->[delay Tb]---+

The product of the present and one-bit-delayed signals, integrated over $T_b$, gives $y\propto\cos(\theta_k-\theta_{k-1})$. No phase change gives $y>0$ (bit 1); a $\pi$ change gives $y<0$ (bit 0). For the example, phase changes no, yes, no, no, yes give 1, 0, 1, 1, 0.

Signal space diagram

Over one bit, the transmitted points are the same as BPSK, $\pm\sqrt{E_b}$ on $\phi_1$; what matters is whether the point stays or moves.

      phase pi          phase 0
 -------*-------+-------*-------> phi1
     -sqrt(Eb)  0   +sqrt(Eb)
   bit 1: stay on same point
   bit 0: jump to other point

Viewed over two bit intervals ($2T_b$), DPSK is binary orthogonal signalling: "no change" $(\sqrt{E_b},\sqrt{E_b})$ and "change" $(\sqrt{E_b},-\sqrt{E_b})$, energy $2E_b$, detected non-coherently.

Differential coding

Differential coding represents each data bit by a change or no change in the encoded sequence, rather than by an absolute level. The encoder compares the present data bit with the previous encoded bit:

$$d_k=\overline{b_k\oplus d_{k-1}}\quad(\text{1: keep previous symbol, 0: toggle})$$

An initial reference bit is chosen arbitrarily. Decoding uses only two neighbouring received bits, $b_k=\overline{d_k\oplus d_{k-1}}$, so a complete inversion of the received stream (e.g. $180^\circ$ phase ambiguity) does not cause errors.

DPSK modulation

Differential PSK = differential encoding + BPSK. The encoded bit $d_k$ selects carrier phase 0 ($d_k=1$) or $\pi$ ($d_k=0$).

 b_k -->[XNOR]--d_k-->[polar level]-->(X)--> DPSK s(t)
          ^       |                    ^
          |  [delay Tb]           A cos wc t
          +--------+

Example, data 1 0 0 1 0 0 1 1 with reference 1:

Each 0 in the data causes a $\pi$ phase change; each 1 leaves the phase unchanged.

DPSK detection

 r(t)->[BPF]-+------------>(X)->[int 0..Tb]->[decide]->b_k
             |              ^                 y>0 -> 1
             +->[delay Tb]--+                 y<0 -> 0

The received bit is multiplied by the previous bit and integrated:

$$y=\int_0^{T_b}A\cos(\omega_ct+\theta_k)\,A\cos(\omega_ct+\theta_{k-1})\,dt\approx\frac{A^2T_b}{2}\cos(\theta_k-\theta_{k-1})$$

The output equals the data. No carrier recovery is needed because any constant phase offset cancels in the difference.

• $P_e=\tfrac12e^{-E_b/N_0}$ (about 1 dB worse than coherent BPSK).
• Errors tend to come in pairs since each bit is the reference for the next.

DPSK (Differential Phase Shift Keying) is a non-coherent form of PSK in which a data bit is represented by the change in carrier phase relative to the previous bit, not by an absolute phase. Typically bit 1 means no phase change and bit 0 means a $180^\circ$ change. The previous bit acts as the phase reference, so the receiver needs no coherent carrier.

Implementation: transmitter

 b_k -->[XNOR]--d_k-->[BPSK modulator]--> DPSK
          ^      |      (phase 0 / pi)
          +-[Tb delay]

1. Differential encoder: $d_k=\overline{b_k\oplus d_{k-1}}$ with an initial reference bit.
2. BPSK modulator: $d_k=1$ gives $A\cos\omega_ct$, $d_k=0$ gives $-A\cos\omega_ct$.

Example: data 1 0 1 1, reference 1 gives $d_k$ = 1 1 0 0 0, phases 0, 0, $\pi$, $\pi$, $\pi$.

Implementation: receiver

 r(t)->[BPF]-+------------>(X)->[int 0..Tb]->[sign]-> b_k
             +->[Tb delay]--^

The present bit is multiplied by the delayed previous bit and integrated; the result is proportional to $\cos(\theta_k-\theta_{k-1})$. Positive means no phase change (1), negative means a change (0). For the example, changes are no, yes, no, no, giving 1 0 1 1.

DPSK is simpler than coherent PSK and immune to phase ambiguity, with $P_e=\tfrac12e^{-E_b/N_0}$.

A modem (MOdulator-DEModulator) is a device that converts digital data from a computer into an analog signal suitable for a telephone line or other analog channel (modulation, using ASK/FSK/PSK/QAM), and converts the received analog signal back into digital data (demodulation).

 [PC]--digital--[Modem]~~analog~~[Modem]--digital--[PC]

Modes of operation

1. Simplex: data flows in one direction only, e.g. a remote sensor sending readings.
2. Half-duplex: both directions, but one at a time; the modems switch between send and receive (line turnaround), e.g. older fax modems.
3. Full-duplex: both directions at the same time, either over a 4-wire line or over a 2-wire line by splitting the band into two frequency channels (e.g. Bell 103: 1070/1270 Hz one way and 2025/2225 Hz the other) or by echo cancellation (V.32).

Modems also work in:

• Asynchronous mode: each character is framed with start and stop bits; no common clock; used for low speeds.
• Synchronous mode: blocks of data are sent with a recovered clock; no start/stop bits, so higher speed and efficiency.