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

sdr.plot.use_style()

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()
sdr.plot.impulse_response(rect, color="k", 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/9a74e69208863d7932070e73d8baab354bb4ad3137da68a31c54c204ad2b8c1d.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()
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.show()

../../_images/655c34a994b8166b0f7b73fc3343ed91b9ecc5ebe8234d7097c85f0f91c5292f.png

plt.figure()
sdr.plot.magnitude_response(rect, sample_rate=sps, color="k", 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/5b6fb4c2b6dba34d7d97c0165962df4057d9fdd825ed0445d9a9dfe02a29f5ea.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()
plt.plot(w, P_rect, color="k", 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.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.show()

../../_images/bbae5aa8121a0fe2d31109cfa72a1ea4a513f338c65fc8173239511068dd0264.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()
sdr.plot.impulse_response(rect, color="k", 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.show()

../../_images/36c0ba41e95879419604f3cb81cd76c1e8eeaeb4727e58b29f2dced1581e7fd8.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()
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.show()

../../_images/a96d5f9b167e4fb26bd81cd7074589aba855c1f6604a361c4b26fa6ebd97e50a.png

plt.figure()
sdr.plot.magnitude_response(rect, sample_rate=sps, color="k", 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.xlabel("Normalized frequency, $f/f_{sym}$")
plt.show()

../../_images/3ad0fb9f62429dc2335c13c92eefa77bc6f4ca2c5b0d91af8d381a3f27c51439.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()
plt.plot(w, P_rect, color="k", 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.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.show()

../../_images/bd893daac98bf79bb975234feb213ddf9262fa7e54933c32b2f76d234bdae613.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()
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/ed0be318d0ef9e44f6d4f5c9cac5758ae5eb090ffe466d431b0191bb7ee90b0f.png

plt.figure()
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.xlabel("Normalized frequency, $f/f_{sym}$")
plt.show()

../../_images/627bb56c99928c73a309a1070535699c8622ac9fb779948d4472e6f04ee65c99.png