Beat frequency sinusoid and its spectrum

Spectrum of a Sinusoid

In this article, we derive the spectrum of a complex sinusoid that acts as the basis for all spectra. In fact, the very definition of the Fourier Transform, whether continuous or discrete, comes from the perspective of a complex sinusoid. Therefore, exploring this derivation will be useful in everything else we learn about DSP.

For an in-depth understanding of complex signals and I/Q processing, you can read the following two articles (the option of downloading them as PDF is available).

Let us start with the continuous-time case.

A Continuous-Time Sinusoid


A continuous-time complex sinusoid with a frequency $f_b$ is defined as
\[
x(t) = e^{j2\pi f_b t} = \cos(2\pi f_bt) + j \sin(2\pi f_bt)
\]

This is known as Euler’s identity. Next, the Continuous Fourier Transform (CFT) is defined as
\[
X(f) = \int _{-\infty}^{+\infty} x(t) e^{-j2\pi f t}
\]

In all real scenarios, this complex sinusoid is limited within some duration, say of $T$ seconds. Therefore, we can write
\[
\begin{aligned}
X(f) &= \int _{-T/2}^{T/2} e^{j2\pi f_b t} e^{-j2\pi f t} \\
&= \int _{-T/2}^{T/2} e^{-j2\pi (f-f_b) t} = \frac{e^{-j2\pi (f-f_b)t}}{-j2\pi (f-f_b)} \Bigg|_{-T/2}^{T/2}
\end{aligned}
\]

because the integral of an exponential function is the exponential function itself, scaled by the constant being multiplied with the variable $t$. After simplification, we get
\[
X(f) = \frac{1}{j2\pi (f-f_b)} \left[e^{j2\pi (f-f_b)\frac{T}{2}} ~-~ e^{-j2\pi (f-f_b)\frac{T}{2}} \right]
\]

The expression in the brackets is $j2\sin(\cdot)$ from Euler’s identity.
\[
\sin \theta = \frac{e^{j\theta} – e^{-j\theta}}{2j}
\]

Consequently, we can write
\[
\begin{aligned}
X(f) &= \frac{\sin\left[2\pi (f-f_b)\frac{T}{2}\right]}{\pi (f-f_b)} = T\frac{\sin\left[\pi (f-f_b)T\right]}{\pi (f-f_b)T}\\
&= T\text{sinc}\left[\pi (f-f_b)T\right] \qquad \text{where}\qquad \text{sinc}(x) = \frac{\sin(x)}{x}
\end{aligned}
\]

Since a sinc function has its peak at zero, we conclude that the Fourier Transform of a complex sinusoid is a sinc function shifted at frequency $f_b$. A limited duration continuous-time sinusoid and its spectrum are shown in the figure below, which is taken from the article on Frequency Modulated Continuous-Wave (FMCW) radar.

Beat frequency sinusoid and its spectrum

Let us explore how the same result can be intuitively derived.

Intuitive Reasoning


Intuitively, the same result can be reached by following the above figure which plots a sinusoid along with its spectrum.

  • An infinitely long complex sinusoid with frequency $f_b$ in time domain has a spectrum that is an impulse at $f_b$.
  • A rectangular signal in time domain is a sinc signal at DC in frequency domain.
  • A limited duration waveform, like the sinusoid above, is similar to an infinite waveform multiplied with a rectangular signal.
  • Multiplication in time domain is convolution in frequency domain.
  • Convolution of the impulse at $f_b$ and the sinc signal at DC just puts the sinc signal at frequency $f_b$, as shown in the spectrum above.

We now turn our attention towards the discrete-time case.

A Discrete-Time Sinusoid


A discrete-time complex sinusoid with a frequency $k_o/N$ is defined as
\[
x[n] = e^{j2\pi \frac{k_o}{N}n} = \cos\left(2\pi \frac{k_o}{N}n\right) + j \sin\left(2\pi \frac{k_o}{N}n\right)
\]

To be consistent with the continuous-time case, the Discrete Fourier Transform (DFT) is defined for a length-$N=2L+1$ sequence as
\[
X[k] = \sum _{n=-L}^{L} x[n] e^{-j2\pi \frac{k}{N}n}
\]

where $k/N$ is the general discrete frequency. As opposed to other books and articles, I write the discrete frequency explicitly as $k/N$ so that one can immediately see the similarity behind $2\pi ft$ and $2\pi (k/N)n$ and recognize the discrete frequency, which loses its meaning when expressed as $2\pi kn/N$.

Now, applying the DFT definition,
\[
X[k] = \sum _{n=-L}^{L} e^{j2\pi \frac{k_o}{N}n} e^{-j2\pi \frac{k}{N}n} = \sum _{n=-L}^{L} e^{-j2\pi \frac{k-k_o}{N}n}
\]

The geometric series formula can be applied here.
\[
\sum _{n=N_1}^{N_2} a^n = \frac{a^{N_1} – a^{N_2+1}}{1-a}
\]

This leads us to the following expression.

\[
X[k] = \frac{e^{j2\pi \frac{k-k_o}{N}L} – e^{-j2\pi \frac{k-k_o}{N}(L+1)}}{1-e^{-j2\pi \frac{k-k_o}{N}}}
\]

Taking $e^{-j\pi\frac{k-k_o}{N}}$ common from both the numerator and the denominator, further simplification of this expression yields
\[
X[k] = \frac{e^{j2\pi \frac{k-k_o}{N}\left(L+\frac{1}{2}\right)} – e^{-j2\pi \frac{k-k_o}{N}\left(L+\frac{1}{2}\right)}}{e^{j\pi \frac{k-k_o}{N}}-e^{-j\pi \frac{k-k_o}{N}}} = \frac{\sin\left[ \pi \frac{k-k_o}{N}\left(2L+1\right)\right]}{\sin\left[\pi \frac{k-k_o}{N}\right]}
\]

This result — known as an aliased sinc function or a Dirichlet Kernel — is similar to the continuous-time case in that the numerator is $2\pi$ times half the temporal support. But it is different in the denominator where we have another $\sin(\cdot)$ function. For large $N$, the denominator approaches the sine argument itself (since $\sin\theta \approx \theta$ for small $\theta$) and becomes much smaller than the numerator. Therefore, it becomes a sinc function as well. So when $\pi k/N$ is small, we get
\[
X[k] \approx \frac{\sin\left[ \pi \frac{k-k_o}{N}\left(2L+1\right)\right]}{\pi \frac{k-k_o}{N}} = (2L+1)\text{sinc}\left[ \pi \frac{k-k_o}{N}\left(2L+1\right)\right]
\]

which corresponds to the continuous-time scenario derived above. In the limiting case, this sinc function reduces to a single impulse.

In relation to this derivation, you might also like the role of symmetry in Fourier Transform.

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