Multilevel modulation significance; QPSK

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.