MATLAB: Frequency mixing

digital signal processing

Why is that if one multiplies a function say g(t) by exp(jwt) in matlab, the frequency resulting signal is upconverted rather for it to be downconverted. mathematically by Fourier analysis; FT-means Fourier transform say g(t)=sin(wt) if FT[g(t)]=G(wo) then FT[g(t)exp(jwt)]= G(wo+w), but in matlab the plotting abs(FT[g(t)exp(jwt)]) gives a result that presents FT[g(t)exp(jwt)]=G(wo-w) Pls, kindly shed light on this!

Best Answer

Let X(\omega) = G(\omega-\omega_0)

Then

X(\omega_0) = G(0)

if \omega_0 is positive, then X(\omega) is shifted to the right of G(\omega)

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MATLAB: How can plot this Periodic Signal with E Amplitude using Matalb and calculate FFT for it.

You’re not calculating the fft but the analytic Fourier transform. To do this, use the definition of the Fourier transform of x(t):

X(w) = int(x(t) * exp(j*w*t), t, T1, T2)

defining w=2*pi*f.

Specifically, with the Symbolic Math Toolbox:

syms E T t w
S1 = int(4*E/T*t * exp(j*w*t), t, 0, T/4);
S2 = int(E * exp(j*w*t), t, T/4, T/2);
S3 = int(-E * exp(j*w*t), t, T/2, 3*T/4);
S4 = int(4*E/T*(t-T) * exp(j*w*t), t, 3*T/4, T);
X(w) = S1 + S2 + S3 + S4;
X(w) = simplify(collect(X(w)))                      % Exponential Expression
X(w) = simplify(collect(rewrite(X(w), 'sincos')))   % Trigonometric Expression
X(w) =
-(4*E - 4*E*exp(T*w*1i) - 4*E*exp((T*w*1i)/4) + 4*E*exp((T*w*3i)/4) + E*T*w*exp((T*w*1i)/2)*2i - E*T*w*exp((T*w*3i)/2)*1i + E*T*w*exp((T*w*3i)/4)*1i)/(T*w^2)
X(w) =
((E*T*cos((3*T*w)/2)*1i + 2*E*T*sin((T*w)/2) - E*T*sin((3*T*w)/2) + E*T*sin((3*T*w)/4) - E*T*cos((3*T*w)/4)*1i - E*T*cos((T*w)/2)*2i)*w + 4*E*cos(T*w) - 4*E + 4*E*cos((T*w)/4) - 4*E*cos((3*T*w)/4) + E*sin(T*w)*4i + E*sin((T*w)/4)*4i - E*sin((3*T*w)/4)*4i)/(T*w^2)

MATLAB: How to find a fourier inverse transform using MATLAB Program

If you have the symbolic toolbox use ifourier() and multiply your function by the Heaviside function to give you an integration for [0 \infty)

   syms t v
   ifourier(exp(-v)*heaviside(v),v)

Keep in mind the convention used here for the Fourier and inverse pair.

The Fourier transform is

 X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-i \omega t} dt

and the inverse is

 x(t) = \dfrac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) e^{i\omega t} d\omega

Best Answer

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MATLAB: How can plot this Periodic Signal with E Amplitude using Matalb and calculate FFT for it.

MATLAB: How to find a fourier inverse transform using MATLAB Program

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