Frequency difference measurements

Question

I am trying to find the frequency difference ($\Delta f= f_1 - f_2$) between two complex sinusoidal signals using Matlab. The two signals are: $$r_1 (t)= A~ e^{j 2 \pi f_1 t}= A \cos(2 \pi f_1 t)+j A\sin(2 \pi f_1 t)$$ $$r_2 (t)= A~ e^{j 2 \pi f_2 t}= A \cos(2 \pi f_2 t)+j A\sin(2 \pi f_2 t)$$ where $f_1=150\, \text{KHz}$ and $f_2=400\, \text{KHz}$

In my Matlab code, I considered the signals are sampled at $F_s=900 \, \text{KHz}$ then

    A=1;
    j=sqrt(-1);
    f_1=150e3;
    f_2=400e3;
    Fs=900e3;    
    n=0:1/Fs:1-1/Fs;
    r1s= A*cos(2*pi*f_1*n)+1j*A*sin(2*pi*f_1*n);
    r2s= A*cos(2*pi*f_2*n)+1j*A*sin(2*pi*f_2*n);

Then, to find $\Delta f$, I multiplied both signals as follows:

     z= r1s.*conj(r2s);
     Fz=fft(z);
     subplot(2,1,1)
     plot(real(Fz));
     subplot(2,1,2)
     plot(imag(Fz));

Then, I applied $\text{FFT}$ and plotted the real and imaginary part of the multiplication. But I could not find the pulse at $f_1 - f_2$. The pulse was at high frequency.

Also, I multiplied the two signals as

     z2= r1s.*r2s;
     Fz2=fft(z2);
     subplot(2,1,1)
     plot(real(Fz2));
     subplot(2,1,2)
     plot(imag(Fz2));

and applied $\text{FFT}$. The pulse was at $f_1 + f_2$, which means that what I am doing makes sense.

My question is why could I not get the frequency difference when I applied the multiplication with complex conjugate?

Answer 1

In the first case you're multiplying $$r_1(t) \times r_2(t) = e^{j\omega_1 t}e^{j\omega_2 t}$$ so in the frequency domain, you're convolving: $$R_1(\omega) \,\circledast \,R_2(\omega) = 2\pi\delta(\omega-\omega_1)\,\circledast\,2\pi\delta(\omega-\omega_2)$$ which gives you a value at $\omega_1 + \omega_2$. In terms of frequency, that's $150 + 400 = 550\, \text{kHz}$

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The Fourier transform of the conjugate is the conjugate of the frequency reversed Fourier Transform: $$r_2(t) \xrightarrow{} R_2(\omega)\\ r_2^{*}(t) \xrightarrow{} R_2^{*}(-\omega)$$ Note that because the Fourier Transform of a complex exponential is real, $R_2(\omega)$ is real, so $$R_2^{*}(-\omega) = R_2(-\omega) = 2\pi\delta((2\pi - \omega) - \omega_2)$$

The point is that $R_2$ is reversed, which in the frequency domain gives you a value at $\omega = 2\pi - \omega_2$.

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With that in mind, in the second case you're multiplying $$r_1(t) \times r_2^{*}(t) = e^{j\omega_1 t}e^{-j\omega_2 t}$$ so in the frequency domain, you're convolving: $$R_1(\omega) \,\circledast \,R_2(-\omega) = 2\pi\delta(\omega-\omega_1)\,\circledast\,2\pi\delta((2\pi-\omega)-\omega_2)$$ which gives you a value in the spectrum at $\omega = \omega_1 + 2\pi - \omega_2$.

In terms of frequency, that's $150 + 900 - 400 = 650\, \text{kHz}$