Pulse shapes¶

import matplotlib.pyplot as plt
import numpy as np
import scipy.signal

import sdr

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

span = 8  # Length of the pulse shape in symbols
sps = 10  # Samples per symbol

Create a rectangular pulse shape for reference.

rect = np.zeros(sps * span + 1)
rect[rect.size // 2 - sps // 2 : rect.size // 2 + sps // 2] = 1 / np.sqrt(sps)

Raised cosine¶

Create three raised cosine pulses with different excess bandwidths. This is achieved using the sdr.raised_cosine() function.

rc_0p1 = sdr.raised_cosine(0.1, span, sps)
rc_0p5 = sdr.raised_cosine(0.5, span, sps)
rc_0p9 = sdr.raised_cosine(0.9, span, sps)

plt.figure(figsize=(10, 5))
sdr.plot.impulse_response(rect, color="k", linestyle=":", label="Rectangular")
sdr.plot.impulse_response(rc_0p1, label=r"$\alpha = 0.1$")
sdr.plot.impulse_response(rc_0p5, label=r"$\alpha = 0.5$")
sdr.plot.impulse_response(rc_0p9, label=r"$\alpha = 0.9$")
plt.show()

../../_images/e82818f4396da7f4bb8ed21293ad391ebb67947a5020c6410a36afa294222e03.png

The raised cosine filter is a Nyquist filter. This means that the impulse response \(h[n]\) is zero at adjacent symbols. Specifically, \(h[n] = 0\) for \(n = \pm k\ T_s / T_{sym}\)

plt.figure(figsize=(10, 5))
sdr.plot.time_domain(np.roll(rc_0p1, -3 * sps))
sdr.plot.time_domain(np.roll(rc_0p1, -2 * sps))
sdr.plot.time_domain(np.roll(rc_0p1, -1 * sps))
sdr.plot.time_domain(np.roll(rc_0p1, 0 * sps))
sdr.plot.time_domain(np.roll(rc_0p1, 1 * sps))
plt.xlim(0, 60)
plt.title("Raised cosine pulses for adjacent symbols")
plt.tight_layout()
plt.show()

../../_images/7203326f9703b91067febbd609566949d71917d78cd1744f5da173080cc61a83.png

plt.figure(figsize=(10, 5))
sdr.plot.magnitude_response(rect, sample_rate=sps, color="k", linestyle=":", label="Rectangular")
sdr.plot.magnitude_response(rc_0p1, sample_rate=sps, label=r"$\alpha = 0.1$")
sdr.plot.magnitude_response(rc_0p5, sample_rate=sps, label=r"$\alpha = 0.5$")
sdr.plot.magnitude_response(rc_0p9, sample_rate=sps, label=r"$\alpha = 0.9$")
plt.xlabel("Normalized frequency, $f/f_{sym}$")
plt.show()

../../_images/3af47d79784ec06e6f3e6c75d2a1018c8c9e8ea7b1e2187a5e91e72d935e7c84.png

Notice the raised cosine pulse with excess bandwidth \(\alpha = 0.1\) has a total bandwidth of nearly \(f_{sym}\). Compare this to \(\alpha = 0.9\), which has a null-to-null bandwidth of nearly \(2 f_{sym}\).

While small \(\alpha\) produces a filter with smaller bandwidth, its side lobes are much higher.

# Compute the one-sided power spectral density of the pulses
w, H_rect = scipy.signal.freqz(rect, 1, worN=1024, whole=False, fs=sps)
w, H_rc_0p1 = scipy.signal.freqz(rc_0p1, 1, worN=1024, whole=False, fs=sps)
w, H_rc_0p5 = scipy.signal.freqz(rc_0p5, 1, worN=1024, whole=False, fs=sps)
w, H_rc_0p9 = scipy.signal.freqz(rc_0p9, 1, worN=1024, whole=False, fs=sps)

# Compute the relative power in the main lobe of the pulses
P_rect = sdr.db(np.cumsum(np.abs(H_rect) ** 2) / np.sum(np.abs(H_rect) ** 2))
P_rc_0p1 = sdr.db(np.cumsum(np.abs(H_rc_0p1) ** 2) / np.sum(np.abs(H_rc_0p1) ** 2))
P_rc_0p5 = sdr.db(np.cumsum(np.abs(H_rc_0p5) ** 2) / np.sum(np.abs(H_rc_0p5) ** 2))
P_rc_0p9 = sdr.db(np.cumsum(np.abs(H_rc_0p9) ** 2) / np.sum(np.abs(H_rc_0p9) ** 2))

plt.figure(figsize=(10, 5))
plt.plot(w, P_rect, color="k", linestyle=":", label="Rectangular")
plt.plot(w, P_rc_0p1, label=r"$\alpha = 0.1$")
plt.plot(w, P_rc_0p5, label=r"$\alpha = 0.5$")
plt.plot(w, P_rc_0p9, label=r"$\alpha = 0.9$")
plt.legend()
plt.xlim(0.25, 1)
plt.ylim(-3, 0)
plt.grid()
plt.xlabel("One-sided normalized frequency, $f/f_{sym}$")
plt.ylabel("Relative power (dB)")
plt.title("Relative power within bandwidths for various raised cosine pulses")
plt.tight_layout()
plt.show()

../../_images/2f5a5158417a6248c06df2121f01426c6036209c1004d3ec604cb87083bb7ac6.png

Square-root raised cosine¶

Create three square-root raised cosine pulses with different excess bandwidths. This is achieved using the sdr.root_raised_cosine() function.

srrc_0p1 = sdr.root_raised_cosine(0.1, span, sps)
srrc_0p5 = sdr.root_raised_cosine(0.5, span, sps)
srrc_0p9 = sdr.root_raised_cosine(0.9, span, sps)

plt.figure(figsize=(10, 5))
sdr.plot.impulse_response(rect, color="k", linestyle=":", label="Rectangular")
sdr.plot.impulse_response(srrc_0p1, label=r"$\alpha = 0.1$")
sdr.plot.impulse_response(srrc_0p5, label=r"$\alpha = 0.5$")
sdr.plot.impulse_response(srrc_0p9, label=r"$\alpha = 0.9$")
plt.legend()
plt.show()

../../_images/83277068aa887a7c9561e854ab3bc5dae38f9acfd353430a202b39c52967db90.png

The square-root raised cosine filter is not a Nyquist filter. Therefore, the impulse response \(h[n]\) is not zero at adjacent symbols.

plt.figure(figsize=(10, 5))
sdr.plot.time_domain(np.roll(srrc_0p1, -3 * sps))
sdr.plot.time_domain(np.roll(srrc_0p1, -2 * sps))
sdr.plot.time_domain(np.roll(srrc_0p1, -1 * sps))
sdr.plot.time_domain(np.roll(srrc_0p1, 0 * sps))
sdr.plot.time_domain(np.roll(srrc_0p1, 1 * sps))
plt.xlim(0, 60)
plt.title("Square-root raised cosine pulses for adjacent symbols")
plt.tight_layout()
plt.show()

../../_images/f228c58166e854c513c8b56cb721c0bcdca10d1fbfcfab39cfdcd89a1d0b63a3.png

plt.figure(figsize=(10, 5))
sdr.plot.magnitude_response(rect, sample_rate=sps, color="k", linestyle=":", label="Rectangular")
sdr.plot.magnitude_response(srrc_0p1, sample_rate=sps, label=r"$\alpha = 0.1$")
sdr.plot.magnitude_response(srrc_0p5, sample_rate=sps, label=r"$\alpha = 0.5$")
sdr.plot.magnitude_response(srrc_0p9, sample_rate=sps, label=r"$\alpha = 0.9$")
plt.legend()
plt.xlabel("Normalized frequency, $f/f_{sym}$")
plt.show()

../../_images/bc3e6f90e3f0de63c9644609d49634361caa786557a36666b9802b4344318eab.png

While the bandwidths of the square-root raised cosine filter are similar to the raised cosine filter, the side lobes are significantly higher. This is due to this filter not being a Nyquist filter.

# Compute the one-sided power spectral density of the pulses
w, H_rect = scipy.signal.freqz(rect, 1, worN=1024, whole=False, fs=sps)
w, H_srrc_0p1 = scipy.signal.freqz(srrc_0p1, 1, worN=1024, whole=False, fs=sps)
w, H_srrc_0p5 = scipy.signal.freqz(srrc_0p5, 1, worN=1024, whole=False, fs=sps)
w, H_srrc_0p9 = scipy.signal.freqz(srrc_0p9, 1, worN=1024, whole=False, fs=sps)

# Compute the relative power in the main lobe of the pulses
P_rect = sdr.db(np.cumsum(np.abs(H_rect) ** 2) / np.sum(np.abs(H_rect) ** 2))
P_srrc_0p1 = sdr.db(np.cumsum(np.abs(H_srrc_0p1) ** 2) / np.sum(np.abs(H_srrc_0p1) ** 2))
P_srrc_0p5 = sdr.db(np.cumsum(np.abs(H_srrc_0p5) ** 2) / np.sum(np.abs(H_srrc_0p5) ** 2))
P_srrc_0p9 = sdr.db(np.cumsum(np.abs(H_srrc_0p9) ** 2) / np.sum(np.abs(H_srrc_0p9) ** 2))

plt.figure(figsize=(10, 5))
plt.plot(w, P_rect, color="k", linestyle=":", label="Rectangular")
plt.plot(w, P_srrc_0p1, label=r"$\alpha = 0.1$")
plt.plot(w, P_srrc_0p5, label=r"$\alpha = 0.5$")
plt.plot(w, P_srrc_0p9, label=r"$\alpha = 0.9$")
plt.legend()
plt.xlim(0.25, 1)
plt.ylim(-3, 0)
plt.grid()
plt.xlabel("One-sided normalized frequency, $f/f_{sym}$")
plt.ylabel("Relative power (dB)")
plt.title("Relative power within bandwidths for various square-root raised cosine pulses")
plt.tight_layout()
plt.show()

../../_images/063e7e7676dedbe508960cc2018ac44432e2d84f5c89121589d22568f84b719d.png

Gaussian¶

Create three raised Gaussian pulses with different time-bandwidth products. This is achieved using the sdr.gaussian() function.

gauss_0p1 = sdr.gaussian(0.1, span, sps)
gauss_0p2 = sdr.gaussian(0.2, span, sps)
gauss_0p3 = sdr.gaussian(0.3, span, sps)

plt.figure(figsize=(10, 5))
sdr.plot.impulse_response(gauss_0p1, label=r"$B T_{sym} = 0.1$")
sdr.plot.impulse_response(gauss_0p2, label=r"$B T_{sym} = 0.2$")
sdr.plot.impulse_response(gauss_0p3, label=r"$B T_{sym} = 0.3$")
plt.show()

../../_images/e3ddbbe9379e1482f7113b6c19110c36a4ee074420ac13f5d24073431421bbba.png

plt.figure(figsize=(10, 5))
sdr.plot.magnitude_response(gauss_0p1, sample_rate=sps, label=r"$B T_{sym} = 0.1$")
sdr.plot.magnitude_response(gauss_0p2, sample_rate=sps, label=r"$B T_{sym} = 0.2$")
sdr.plot.magnitude_response(gauss_0p3, sample_rate=sps, label=r"$B T_{sym} = 0.3$")
plt.legend()
plt.xlabel("Normalized frequency, $f/f_{sym}$")
plt.show()

../../_images/70a19a319ee55a061f0854b0b58a1c76e7bcc99b1933e23367981199350524b8.png

Last update: Dec 06, 2023