convolution([1,5,6], [3,2])
[3, 17, 28, 12] [3, 17, 28, 12] |
(0.79999999999999982, 1.0, 0.0) (0.79999999999999982, 1.0, 0.0) |
#Plot a Lissajous Curve (http://en.wikipedia.org/wiki/Lissajous_curve)
def parametricFuncs(f):
def x(t):
return cos(2*pi*t)
def y(t):
return sin(2*f*pi*t)
return (x, y)
N = 5
g_lissajours = [parametric_plot((parametricFuncs(f)), (0, 1.0), color=hue(f * (0.99 / N)))for f in range(1, N)]
comb = None
for g in g_lissajours:
if (comb == None):
comb = g
else:
comb = comb + g
comb.show(aspect_ratio = 1.0)
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#QPSK - a simuation of QPSK moduling
def randomStepVal(amp=1.0):
if (random() >= 0.5):
return amp
else:
return 0
#given a list of tuple, and generate a piecewise function
def PiecewiseWithConsts(lst):
#Gather all const values, and generate a dict for it
constFuncs = dict()
for pair in lst:
if (not pair[1] in constFuncs):
constval = pair[1]
constFunc(x) = constval
constFuncs[pair[1]] = constFunc
funclist = [[(lst[0][0], lst[0][0]), constFuncs[lst[0][1]]]]
for pair in lst:
oldpair = funclist[-1][0];
funclist[-1][0] = (oldpair[0], pair[0])
funclist.append([(pair[0], pair[0] + 1), constFuncs[pair[1]]])
return Piecewise(funclist)
def randomStepSignal(T):
valdict = dict()
def f_stepSig(t):
smpT = floor(t / T) * T
if (smpT in valdict):
return valdict[smpT];
else:
valdict[smpT] = randomStepVal()
return valdict[smpT]
return f_stepSig
#IQ moduling
def module_IQ(mapped, w):
(f_I, f_Q) = mapped
def f_IQMod(t):
return f_I(t) * cos (w * t) - f_Q(t) * sin(w*t)
return f_IQMod
def mapBoolToValue(boolExp, trueValue, falseValue):
if (boolExp):
return trueValue
else:
return falseValue
def xor(a, b):
return int(a).__xor__(int(b))
def combSignalBits(sig, period, t, nBits):
val = 0;
sampT = floor(t / (period * nBits)) * (period * nBits) + period / 2.0
for i in range(nBits):
val = val | (Integer(sig(sampT + period * i)) << i)
return val
def IQMapper_ByList(valMap, indexer):
def f_I(t):
return valMap[indexer(t)][0]
def f_Q(t):
return valMap[indexer(t)][1]
return (f_I, f_Q)
#The QPSK mapper, give a signal function and it's period, return a
#The mapper using Gray Code to arrange values
def IQMapper_qpsk(sig, period):
amp = 1 / sqrt(2)
valueMap = [(amp, amp), (-amp, amp), (-amp, -amp), (amp, -amp)]
def combValue(t):
return combSignalBits(sig, period, t, 2)
return IQMapper_ByList(valueMap, combValue)
def IQMapper_8psk(sig, period):
C = cos(pi / 8)
S = sin(pi / 8)
valueMap = [(C, S), (S, C), (-S, C), (-C, S), (-C, -S), (-S, -C), (S, -C), (C, -S)]
def combValue(t):
return combSignalBits(sig, period, t, 3)
return IQMapper_ByList(valueMap, combValue)
def IQMapper_qam(sig, period):
A = 1.0 / sqrt(2.0)
A3 = A * 3.0
valueMap = [(A3, A3), (A, A3), (-A, A3), (-A3, A3), (-A3, A), (-A, A), (A, A), (A3, A),
(A3, -A), (A, -A), (-A, -A), (-A3, -A), (-A3, -A3), (-A, -A3), (A, -A3), (A3, -A3)]
def combValue(t):
return combSignalBits(sig, period, t, 4)
return IQMapper_ByList(valueMap, combValue)
def moduleSignal(sig, period, mapper, moduler, w = 0):
def sig_wrap(t):
return sig(t)
mapped = mapper(sig_wrap, period)
if (w == 0):
w = 2 * pi / period;
moduled = moduler(mapped, w)
lst = [sig_wrap]
for mp in mapped:
lst.append(mp)
lst.append(moduled)
return lst
T = 0.5
count = 64
sig = randomStepSignal(T)
plotFuncs = moduleSignal(sig, T, IQMapper_qpsk, module_IQ)
plotRange = (0.0, T * count)
plotFuncs += moduleSignal(sig, T, IQMapper_8psk, module_IQ)
plotFuncs += moduleSignal(sig, T, IQMapper_qam, module_IQ)
graphs = [plot(func, plotRange) for func in plotFuncs]
for g in graphs:
g.show(aspect_ratio = 1.0)
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sinc(x) = sin(pi * x) / (pi * x)
plot(sinc, x, -10, 10).show(ymin=-1)
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#Create a Band pass filter
def createBandFilter(f, B, Q = 1.0):
h(t) = sinc(2 * B * t) * cos(2 * pi * Q * B * t) / (1 - (4 * Q * B * t)^2)
return h
h = createBandFilter(0, 0.5, 0.9)
plot (h, (-3.0, 3.0), aspect_ratio = 1.0)
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funcs = moduleSignal(sig, T, IQMapper_qpsk, module_IQ)
graphs = [plot(func, (0.0, T * 32)) for func in funcs]
for g in graphs:
g.show(aspect_ratio = 1.0)
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A = Matrix([[1 * exp(2 * pi * i), -2], [3, 4]])
w = vector([1, 2 * i])
w * A
(6*I + 1, 8*I - 2) (6*I + 1, 8*I - 2) |
w * 0.5
(0.500000000000000, 1.00000000000000*I) (0.500000000000000, 1.00000000000000*I) |
len(w)
2 2 |
range(10)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] |
#generate a n * n transform matrix for DFT operation
def fourierTransformMatrix(n):
matlist = []
for j in range(n):
linelst = []
for k in range(n):
linelst.append(exp(- (i * 2 * pi * j * k) / n))
matlist.append(linelst)
return Matrix(matlist)
#Input time domain list of values, return Freq Domain list
def dft(numlist):
vec = vector([num for num in numlist])
n = len(vec)
return [CC(val / n) for val in (vec * fourierTransformMatrix(n))]
def phase(complexNum):
ccnum = CC(complexNum)
absval = abs(ccnum)
if (absval > 0.00001):
return acos(ccnum.real() / absval)
else:
return 0.0
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freq = 50 #50Hz min frequency wave, T = 0.01s
sig(t) = cos((4 * freq) * (2 * pi * t)) + 3 * sin((2 * freq) * (2 * pi * t)) + 2 * cos( (3 *freq) * (2 * pi * t)) + sin(freq * (2 * pi * t)) + cos(freq * (2 * pi * t))
maxFreq = freq * 4
smpInterval = 1 / maxFreq / 2.5 #0.01s per sample
nSamples = 50
#smpInterval = 1.0 / freq / nSamples #Sample only one period
samples = [sig(x * smpInterval) for x in range(nSamples)]
timeDomain = list_plot(samples)
freqVec = dft(samples)
freqDomainAmp = list_plot([abs(fd) for fd in freqVec], hue=0.1)
freqDomainPhase = list_plot([phase(fd) for fd in freqVec], hue = 0.4)
show(timeDomain, aspect_ratio = 3)
show(freqDomainAmp, legend_title="Amp", aspect_ratio = 5)
show(freqDomainPhase, legend_title="Phase", aspect_ratio = 5)
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exp(- i * pi / 2)
-I -I |
phase(CC(1 + 2 * i))
Traceback (click to the left of this block for traceback) ... NameError: name 'phase' is not defined Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_31.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cGhhc2UoQ0MoMSArIDIgKiBpKSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmp_o4Phy/___code___.py", line 3, in <module>
exec compile(u'phase(CC(_sage_const_1 + _sage_const_2 * i))
File "", line 1, in <module>
NameError: name 'phase' is not defined
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help(phase)
Traceback (click to the left of this block for traceback) ... NameError: name 'phase' is not defined Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_34.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("aGVscChwaGFzZSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpMWgfIL/___code___.py", line 2, in <module>
exec compile(u'help(phase)
File "", line 1, in <module>
NameError: name 'phase' is not defined
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