Solution of Exercise 6

3 files changed, 224 insertions, 0 deletions

diff --git a/exercise06/helpers/ex_6_1.py b/exercise06/helpers/ex_6_1.py
new file mode 100644
index 0000000..2ba7096
--- /dev/null
+++ b/exercise06/helpers/ex_6_1.py
@@ -0,0 +1,69 @@
+#!/usr/bin/env python3
+
+# SPDX-License-Identifier: BSD-3-Clause
+#
+# Copyright 2020 Philipp Le
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above copyright
+#    notice, this list of conditions and the following disclaimer in the
+#    documentation and/or other materials provided with the distribution.
+#
+# 3. Neither the name of the copyright holder nor the names of its
+#    contributors may be used to endorse or promote products derived from
+#    this software without specific prior written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
+# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
+# THE POSSIBILITY OF SUCH DAMAGE.
+
+import numpy
+import scipy.signal
+import matplotlib.pyplot as plt
+
+b = [2, -1]
+a = [1, -0.5]
+
+w, h = scipy.signal.freqz(b, a, 8)
+
+print(scipy.signal.tf2zpk(b,a))
+
+w = numpy.round(w*100)/100
+h = numpy.round(h*100)/100
+
+print("$\\phi$ in \\si{rad} & $\\left|\\underline{H}\\left(e^{j\\phi}\\right)\\right|$ & $\\arg\\left(\\underline{H}\\left(e^{j\\phi}\\right)\\right)$ in \\si{rad} \\\\")
+for n in range(len(w)):
+    print("$%g$ & $%g$ & $%g$ \\\\" % (w[n], abs(h[n]), (numpy.angle(h[n]))))
+    
+print()
+
+elems = []
+for n in range(len(w)):
+    elems.append("(%g, %g)" % (w[n], abs(h[n])))
+print("coordinates {%s}" % " ".join(elems))
+
+elems = []
+for n in range(len(w)):
+    elems.append("(%g, %g)" % (w[n], (numpy.angle(h[n]))))
+print("coordinates {%s}" % " ".join(elems))
+
+fig = plt.figure()
+ax1 = fig.add_subplot(1, 1, 1)
+ax1.plot(w, abs(h), 'b')
+ax2 = ax1.twinx()
+ax2.plot(w, (numpy.angle(h)), 'g')
+plt.show()
diff --git a/exercise06/helpers/ex_6_2.py b/exercise06/helpers/ex_6_2.py
new file mode 100644
index 0000000..dcbfc7a
--- /dev/null
+++ b/exercise06/helpers/ex_6_2.py
@@ -0,0 +1,80 @@
+#!/usr/bin/env python3
+
+# SPDX-License-Identifier: BSD-3-Clause
+#
+# Copyright 2020 Philipp Le
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above copyright
+#    notice, this list of conditions and the following disclaimer in the
+#    documentation and/or other materials provided with the distribution.
+#
+# 3. Neither the name of the copyright holder nor the names of its
+#    contributors may be used to endorse or promote products derived from
+#    this software without specific prior written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
+# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
+# THE POSSIBILITY OF SUCH DAMAGE.
+
+import numpy
+import scipy.signal
+import matplotlib.pyplot as plt
+
+b = [1, 0.5+1j, 2]
+a = [1]
+
+w, h = scipy.signal.freqz(b, a, 16, whole=True)
+
+print(scipy.signal.tf2zpk(b,a))
+
+w = 1e6*w/numpy.pi
+
+w = numpy.round(w*100)/100
+h = numpy.round(h*100)/100
+
+print("$f$ in \\si{kHz} & $\\left|\\underline{H}\\left(e^{j 2\\pi f T_S}\\right)\\right|$ & $\\arg\\left(\\underline{H}\\left(e^{j 2\\pi f T_S}\\right)\\right)$ in \\si{rad} \\\\")
+for n in range(len(w)):
+    if w[n] <= 1e6:
+        print("$%g$ & $%g$ & $%g$ \\\\" % (w[n]/1000, abs(h[n]), (numpy.angle(h[n]))))
+    else:
+        print("$%g$ & $%g$ & $%g$ \\\\" % (w[n]/1000-2e3, abs(h[n]), (numpy.angle(h[n]))))
+    
+print()
+
+elems = []
+for n in range(len(w)):
+    if w[n] <= 1e6:
+        elems.append("(%g, %g)" % (w[n], abs(h[n])))
+    else:
+        elems.append("(%g, %g)" % (w[n]-2e6, abs(h[n])))
+print("coordinates {%s}" % " ".join(elems))
+
+elems = []
+for n in range(len(w)):
+    if w[n] <= 1e6:
+        elems.append("(%g, %g)" % (w[n], (numpy.angle(h[n]))))
+    else:
+        elems.append("(%g, %g)" % (w[n]-2e6, (numpy.angle(h[n]))))
+print("coordinates {%s}" % " ".join(elems))
+
+fig = plt.figure()
+ax1 = fig.add_subplot(1, 1, 1)
+ax1.plot(w, abs(h), 'b')
+ax2 = ax1.twinx()
+ax2.plot(w, (numpy.angle(h)), 'g')
+plt.show()
diff --git a/exercise06/helpers/ex_6_4.py b/exercise06/helpers/ex_6_4.py
new file mode 100644
index 0000000..0e1bb9e
--- /dev/null
+++ b/exercise06/helpers/ex_6_4.py
@@ -0,0 +1,75 @@
+#!/usr/bin/env python3
+
+# SPDX-License-Identifier: BSD-3-Clause
+#
+# Copyright 2020 Philipp Le
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above copyright
+#    notice, this list of conditions and the following disclaimer in the
+#    documentation and/or other materials provided with the distribution.
+#
+# 3. Neither the name of the copyright holder nor the names of its
+#    contributors may be used to endorse or promote products derived from
+#    this software without specific prior written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
+# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
+# THE POSSIBILITY OF SUCH DAMAGE.
+
+import numpy
+
+x = numpy.array([-0.5, 1, -2, 2])
+
+print("TEST ================================")
+
+print(numpy.fft.fft(x))
+
+print("a) ==================================")
+
+for k in range(4):
+    eq_str = "\\underline{X}[%d] = " % (k)
+    elems = []
+    res = 0
+    for n in range(4):
+        #elems.append("\\underline{x}[%d] \\cdot e^{-j \\frac{2\\pi}{N} %d \\cdot %d}" % (n, k, n))
+        elems.append("\\underline{x}[%d] \\cdot e^{-j \\frac{\\pi}{2} %d \\cdot %d}" % (n, k, n))
+        res += x[n] * numpy.exp(-1j*(numpy.pi/2)*n*k)
+    eq_str += " + ".join(elems)
+    eq_str += " = " + str(res)
+    print (eq_str)
+
+print("d) ==================================")
+
+E = numpy.zeros(2)
+O = numpy.zeros(2)
+for k in range(2):
+    eq_str = "\\underline{E}[%d] = \\underline{x}[0] + \\underline{w}_2^{%d} \\cdot \\underline{x}[2]" % (k, k)
+    E[k] = x[0] + numpy.exp(-1j*(numpy.pi)*k) * x[2]
+    eq_str += " = " + str(E[k])
+    print (eq_str)
+    
+    eq_str = "\\underline{O}[%d] = \\underline{x}[1] + \\underline{w}_2^{%d} \\cdot \\underline{x}[3]" % (k, k)
+    O[k] = x[1] + numpy.exp(-1j*(numpy.pi)*k) * x[3]
+    eq_str += " = " + str(O[k])
+    print (eq_str)
+
+for k in range(4):
+    eq_str = "\\underline{X}[%d] = \\underline{E}[%d] + \\underline{w}_2^{%d} \\cdot \\underline{O}[%d]" % (k, (k%2), k, (k%2))
+    res = E[k%2] + numpy.exp(-1j*(numpy.pi/2)*k) * O[k%2]
+    eq_str += " = " + str(res)
+    print (eq_str)