TX Pulse Shaping & Matched Filters

Est. read time: 1 minute | Last updated: August 29, 2024 by John Gentile

Contents

References

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
from rfproto import filter, modulation, plot

The why (bandwidth and power amplifiers) -> https://dsp.stackexchange.com/questions/41130/envelope-behavior-difference-between-qpsk-oqpsk-and-pi-4-qpsk
https://en.wikipedia.org/wiki/Raised-cosine_filter

# CCSDS OQPSK SRRC rolloff=0.5: https://public.ccsds.org/Pubs/413x0g3e1.pdf
rrc_test = filter.RootRaisedCosine(17.225e6, 7.5e6, 0.5, 63)
# The matched filter is a time-reversed and conjugated version of the signal
# NOTE: this is moot for a uniform, real filter...
rrc_mf = np.conj(rrc_test[::-1])
plot.filter_coefficients(rrc_mf)
plt.show()

plot.filter_response(rrc_mf)
plt.show()

png

m = 4
qpsk_mod = modulation.MPSKModulation(m)

# simulate random binary input values
num_symbols = 2400
num_disp_sym = 16
rand_symbols = np.random.randint(0, m, num_symbols)
qpsk_symbols = qpsk_mod.modulate(rand_symbols)

L = 4  # Upsample ratio (Samples per Symbol)

def pulse_shape(symbols: np.ndarray, OSR: int, h: np.ndarray):
    """ Similar to np.convolve(x, h, 'same') """
    N = len(h) # filter length (== number of symbols in filter impulse response)
    sym_prepend = np.insert(symbols, 0, symbols[0]*np.ones(N // 2))
    syms = np.append(sym_prepend, symbols[-1]*np.ones(N // 2))

    # zero-insert upsampling
    tx = np.zeros(len(syms) * OSR, dtype=complex)
    tx[::OSR] = syms

    # apply pulse shape filter using direct-form FIR SciPy convolution
    conv_out = signal.lfilter(h, 1, tx)

    # truncate first samples due to prepend and apped to align output with input
    return conv_out[N * OSR:]
 
rc_test = filter.RootRaisedCosine(L, 1, 0.25, 45)
qpsk_tx_filtered = pulse_shape(qpsk_symbols, L, rc_test)

fig, ax = plt.subplots()
ax.plot(np.real(qpsk_tx_filtered[:num_disp_sym * L]), '.-', label='Pulse shaped output')
num_taps = 64
for i in range(num_disp_sym):
    if not i:
        plt.plot([i*L,i*L], [0, np.real(qpsk_tx_filtered[i*L])], color='k', label='Symbol')
    else:
        plt.plot([i*L,i*L], [0, np.real(qpsk_tx_filtered[i*L])], color='k')
plt.grid(True)
plt.legend()
plt.show()

png

_,_ = plot.eye(qpsk_tx_filtered.real, L)

png

plot.spec_an(qpsk_tx_filtered, fs=L*1e6, fft_shift=True, show_SFDR=False, y_unit="dB")
plt.show()

png

# Pass transmitted waveform through same RRC (matched filter)
rx_shaped = signal.lfilter(rc_test, 1, qpsk_tx_filtered)
# don't plot begining samples while starting filter convolution process
transient = (len(rc_test)//2 + 1) * L
_,_ = plot.eye(rx_shaped.real[transient:], L )

# adjust for best EVM, similar to slicer
timing_offset = 4
#plot.IQ(qpsk_tx_filtered[transient + timing_offset::4], alpha=0.3)
plot.IQ(rx_shaped[transient + timing_offset::4], alpha=0.3)
plt.show()

png

TX Pulse Shaping & Matched Filters

References