Validity of differentiation property of Fourier transform

Question

we know the differentiation property of Fourier transform says that, if $$x(t)\longleftrightarrow X(j\omega)$$ then $$\dfrac{d}{dt}x(t)\longleftrightarrow j\omega X(j\omega)$$ We know that we can use this property to find Fourier transform of signum function $sgt(t)$, but cannot be used to find Fourier transform of unit step function $u(t)$.

My argument with intuition is that, we can apply this property only if the average value of the signal is zero. That is, $$\lim_{T_1\to \infty}\frac{1}{2T_1}\int_{-T_1}^{T_1}x(t)dt=0$$

Is this correct or any other validation exists?

Thanks in advance.

Answer 1

You're right about your intuition, but note that the differentiation property is always valid. This can be seen as follows:

If $x(t)$ is a real-valued function which is constant everywhere except at $t=0$ where it has a discontinuity such that $x(0^+)-x(0^-)=1$, then you have

$$\frac{dx(t)}{dt}=\delta(t)\tag{1}$$

In the frequency domain, this is equivalent to

$$j\omega X(j\omega)=1\tag{2}$$

From $(2)$ we can conclude that

$$X(j\omega)=\frac{1}{j\omega}+c\delta(\omega),\quad c\in\mathbb{R}\tag{3}$$

because $j\omega\delta(\omega)=0$.

Note that the real part of $X(j\omega)$ corresponds to the even part of $x(t)$, and the imaginary part corresponds to the odd part of $x(t)$. For the given $x(t)$, the even part is just its DC value. So if $x_0$ is the DC value of $x(t)$, i.e.,

$$x_0=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}x(t)dt\tag{4}$$

then we must have

$$\mathcal{F}\{x_0\}=2\pi x_0\delta(\omega)=c\delta(\omega)\tag{5}$$

from which we obtain

$$c=2\pi x_0\tag{6}$$

Consequently, the Fourier transform of $x(t)$ is given by

$$X(j\omega)=\frac{1}{j\omega}+2\pi x_0\delta(\omega)\tag{7}$$

For $x(t)=\frac12\textrm{sgn}(t)$ we have $x_0=0$ and consequently,

$$\mathcal{F}\left\{\frac12\textrm{sgn}(t)\right\}=\frac{1}{j\omega}\tag{8}$$

and for $x(t)=u(t)$ we have $x_0=\frac12$ and

$$\mathcal{F}\left\{u(t)\right\}=\frac{1}{j\omega}+\pi\delta(\omega)\tag{9}$$

Also take a look at this and this question, which deal with the same problem in discrete time.