Phase-shift keying¶

import matplotlib.pyplot as plt
import numpy as np

import sdr

%config InlineBackend.print_figure_kwargs = {"facecolor" : "w"}
%matplotlib inline
# %matplotlib widget

sdr.plot.use_style()

In the sdr library, phase-shift keying modulation is available in the sdr.PSK class.

def analyze_psk(psk, esn0):
    # Generate random decimal symbols
    s = np.random.randint(0, psk.order, 100_000)

    # Modulate decimal symbols to complex symbols
    x = psk.map_symbols(s)

    # Add AWGN to complex symbols to achieve desired Es/N0
    snr = sdr.esn0_to_snr(esn0, sps=1)
    x_hat = sdr.awgn(x, snr)

    plt.figure()
    plt.subplot(1, 2, 1)
    sdr.plot.symbol_map(psk.symbol_map, limits=(-2, 2))
    plt.subplot(1, 2, 2)
    sdr.plot.constellation(x_hat, persistence=True, bins=150, limits=(-2, 2))
    plt.title(f"Constellation at {esn0} dB $E_s/N_0$")
    plt.suptitle(f"{psk.order}-PSK constellation")
    plt.show()

    y = psk.modulate(s)
    # h_srrc = sdr.root_raised_cosine(0.1, 6, sps)
    # tx_mf = sdr.Interpolator(sps, h_srrc)
    # y = tx_mf(x)

    plt.figure()
    sdr.plot.time_domain(y[0:1000])
    plt.show()

Constellations¶

BPSK¶

bpsk = sdr.PSK(2, sps=10, pulse_shape="srrc")
analyze_psk(bpsk, 6)

../../_images/657f2a556437360a147abaab06042159685bc895c3df8344f8ece9eff3eba9b0.png

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QPSK¶

qpsk = sdr.PSK(4, phase_offset=45, sps=10, pulse_shape="srrc")
analyze_psk(qpsk, 9)

../../_images/d16e1bf2769600d3829e291742cc61587eab18a3387b72ea8297b7e185fd8bed.png

../../_images/abc640b4eb3596be14852e3d248010e77582d07f35c307efb8b835c260fc7b3c.png

8-PSK¶

psk8 = sdr.PSK(8, sps=10, pulse_shape="srrc")
analyze_psk(psk8, 12)

../../_images/ce2e9b25fc204b54c0af09f4ac8dcf6dfcb1e67671bc165f8720cdac887a0803.png

../../_images/2e4437f79edfeaf8768814592d5c49e486b5f119603b49d2e7697e1e7b48ca2c.png

16-PSK¶

psk16 = sdr.PSK(16, sps=10, pulse_shape="srrc")
analyze_psk(psk16, 18)

../../_images/6e1e001eff7798136acb358747c7b6217d991c970157a6dda99d2f8e84808ef8.png

../../_images/779e684e6847c9a63d37e276692021b9457c42435b2352559a4fa66866f010fe.png

Error rate curves¶

def error_rates(psk, ebn0):
    esn0 = sdr.ebn0_to_esn0(ebn0, psk.bps)
    snr = sdr.esn0_to_snr(esn0)

    ber = sdr.ErrorRate()
    ser = sdr.ErrorRate()

    for i in range(snr.size):
        s = np.random.randint(0, psk.order, int(1e6))
        a = psk.map_symbols(s)
        a_tilde = sdr.awgn(a, snr[i])
        s_hat, a_hat = psk.decide_symbols(a_tilde)

        ber.add(ebn0[i], sdr.unpack(s, psk.bps), sdr.unpack(s_hat, psk.bps))
        ser.add(esn0[i], s, s_hat)

    return ber, ser

ebn0 = np.linspace(-2, 10, 20)

bpsk_ber, bpsk_ser = error_rates(bpsk, ebn0)
qpsk_ber, qpsk_ser = error_rates(qpsk, ebn0)
psk8_ber, psk8_ser = error_rates(psk8, ebn0)
psk16_ber, psk16_ser = error_rates(psk16, ebn0)

Bit error rate curves¶

plt.figure()
ebn0 = np.linspace(-2, 10, 200)
sdr.plot.ber(ebn0, bpsk.ber(ebn0), label="BPSK theoretical")
sdr.plot.ber(ebn0, qpsk.ber(ebn0), label="QPSK theoretical")
sdr.plot.ber(ebn0, psk8.ber(ebn0), label="8-PSK theoretical")
sdr.plot.ber(ebn0, psk16.ber(ebn0), label="16-PSK theoretical")
plt.gca().set_prop_cycle(None)
sdr.plot.ber(*bpsk_ber.error_rates(), linestyle="none", marker=".", label="BPSK simulated")
sdr.plot.ber(*qpsk_ber.error_rates(), linestyle="none", marker=".", label="QPSK simulated")
sdr.plot.ber(*psk8_ber.error_rates(), linestyle="none", marker=".", label="8-PSK simulated")
sdr.plot.ber(*psk16_ber.error_rates(), linestyle="none", marker=".", label="16-PSK simulated")
plt.ylim(1e-6, 1e0)
plt.title("Bit error rate curves for PSK modulation in AWGN")
plt.show()

../../_images/d5fd2181506d76db21775d8f15d2ba0f7652a9add5bd4f2649c653eb44f59ca4.png

Symbol error rate curves¶

plt.figure()
esn0 = np.linspace(-4, 16, 200)
sdr.plot.ser(esn0, bpsk.ser(esn0), label="BPSK theoretical")
sdr.plot.ser(esn0, qpsk.ser(esn0), label="QPSK theoretical")
sdr.plot.ser(esn0, psk8.ser(esn0), label="8-PSK theoretical")
sdr.plot.ser(esn0, psk16.ser(esn0), label="16-PSK theoretical")
plt.gca().set_prop_cycle(None)
sdr.plot.ser(*bpsk_ser.error_rates(), linestyle="none", marker=".", label="BPSK simulated")
sdr.plot.ser(*qpsk_ser.error_rates(), linestyle="none", marker=".", label="QPSK simulated")
sdr.plot.ser(*psk8_ser.error_rates(), linestyle="none", marker=".", label="8-PSK simulated")
sdr.plot.ser(*psk16_ser.error_rates(), linestyle="none", marker=".", label="16-PSK simulated")
plt.ylim(1e-6, 1e0)
plt.title("Symbol error rate curves for PSK modulation in AWGN")
plt.show()

../../_images/bf153fde0284777e91f218547dc92ed75c808737aab03b225071905f568abcb8.png

Symbol mapping¶

The mapping of decimal symbols to complex symbols is important. Ideally, adjacent symbol errors only result in 1 bit error. This is generally accomplished using a Gray code (the default in :class:sdr.PSK).

psk8_bin = sdr.PSK(8, symbol_labels="bin")
psk8_gray = sdr.PSK(8, symbol_labels="gray")

plt.figure()
plt.subplot(1, 2, 1)
sdr.plot.symbol_map(psk8_bin.symbol_map, limits=(-2, 2))
plt.title(f"Symbol map for binary coded 8-PSK")
plt.subplot(1, 2, 2)
sdr.plot.symbol_map(psk8_gray.symbol_map, limits=(-2, 2))
plt.title(f"Symbol map for gray coded 8-PSK")
plt.show()

../../_images/ecd82c10feadc6bb30aa8d8117187c849f94f2b464a8daf8c9d817adaef8f8f1.png

Since adjacent symbol errors can lead to multiple bit errors, binary-coded 8-PSK has worse bit error performance.

plt.figure()
ebn0 = np.linspace(-2, 10, 200)
sdr.plot.ber(ebn0, sdr.PSK(8, symbol_labels="bin").ber(ebn0), label="8-PSK w/ binary code")
sdr.plot.ber(ebn0, sdr.PSK(8, symbol_labels="gray").ber(ebn0), label="8-PSK w/ gray code")
plt.title("Bit error rate curves for 8-PSK modulation in AWGN")
plt.show()

../../_images/17223cd7431436fdeffe2a43c7caeaacad2f0b2497872bb8923caba545b4ed54.png