Pulse shapes¶
import matplotlib.pyplot as plt
import numpy as np
import scipy.signal
import sdr
%config InlineBackend.print_figure_kwargs = {"facecolor" : "w"}
# %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()
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()
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.xlim(-3, 3)
plt.ylim(-70, 20)
plt.xlabel("Normalized frequency, $f/f_{sym}$")
plt.show()
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()
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()
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()
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.xlim(-3, 3)
plt.ylim(-70, 20)
plt.xlabel("Normalized frequency, $f/f_{sym}$")
plt.show()
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()
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()
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()
Last update:
Aug 19, 2023