3.2 The First Element: Oscillators

In [2]: 
def plotspec(x, Ts):
    fig = figure()
    ax1 = fig.add_subplot(211)
    ax1.plot(x)
    
    q = fft.fft(x)
    ax2 = fig.add_subplot(212)
    ax2.plot(fft.fftfreq(len(x), Ts), abs(q))
    
def arbspec(s, time, Ts):
    t = linspace(0.0, time, time/Ts)
    x = s(t)
    plotspec(x, Ts)
    
def speccos(f, phi, time, Ts):
    def s(t):
        return cos(2*pi*f*t+phi)
    arbspec(s, time, Ts)

3.5. Mimic the code in speccos.m to find the spectrum of a cosine wave

  1. Of different frequencies, f=1,2,20,30Hz
In [3]: 
for f in [1,2,20,30]:
    speccos(f, 0, 2.0, 1.0/100.0)
  1. For different phases, ph=0,0.1,pi/8,pi/2 rad
In [4]: 
for phi in [0,0.1,pi/8.0,pi/2.0]:
    speccos(10, phi, 2.0, 1.0/100.0)

For different sampling rates, Ts=1/10,1/1k, 1/10k

In [5]: 
for Fs in [10,1000,10000]:
    speccos(10, 0, 2.0, 1.0/float(Fs))

3.6. Let x1(t) be a cosine wave of frequency f = 10, x2(t) be of f=18 and x3 be of f=33. Let x(t) = x1+0.5x2+2x3. Find the spectrum of x. What property of the FT does this illustrate?

In [6]: 
def cosf(f, b):
    def s(t):
        return b*cos(2*pi*f*t)
    return s
    
def xt(t):
    x1 = cosf(10, 1.0)
    x2 = cosf(18, 0.5)
    x3 = cosf(33, 2.0)
    
    return x1(t) + x2(t) + x3(t)

arbspec(xt, 2.0, 1.0/100.0)

3.7. Find the spectrum of a cosine wave when:

  1. phi is a function of time. Try phi(t) = 10pit
  2. phi(t) = pi*t**2
  3. f is a function of time. Try f(t) = sin(2pit)
  4. f(t) = t**2
In [7]: 
def osccos(phit, ft):
    def s(t):
        return cos(2.0*pi*ft(t)*t + phit(t))
    return s

def ocspec(phit, ft):
    arbspec(osccos(phit, ft), 2.0, 1.0/100.0)
    
ocspec(lambda t: 10*pi*t, lambda t: linspace(10,10,len(t)))
ocspec(lambda t: pi*(t**2), lambda t: linspace(10,10,len(t)))
ocspec(lambda t: linspace(0,0,len(t)), lambda t: sin(2*pi*t))
ocspec(lambda t: linspace(0,0,len(t)), lambda t: t**2)
In [7]: