fourier transform - Difference between DC component and zero frequency component of signal

We know that Fourier Transform of a signal exists if it is absolutely integrable and it exists for periodic signals if impulse functions are allowed.

If we consider the fourier transform of $\text{rect}(t)$ , we get $\text{sinc}(f)$ in frequency domain. Which has got zero frequency component as $1$. But we all know, DC value of $\text{rect}(t)$ is zero.

My question is:

1. If a signal has got a zero frequency component in frequency domain ,There must be DC value in time domain. But why there is no DC value in case of $\text{rect}(t)$ in time domain?


2. What is the difference between DC component and zero frequency component?

Answer

Let's first have a look at the rectangular signal given as an example in your question. If you have a rectangle $s(t)$ in the time domain which is $1$ in the interval $[-T/2,T/2]$ and zero elsewhere, its Fourier transform is $S(f)=T\text{sinc}(Tf)$, where I use $\text{sinc}(x)=\sin(\pi x)/(\pi x)$. The value of its Fourier transform at $f=0$ equals $S(0)=T$, which corresponds to

$$\int_{-\infty}^{\infty}s(t)dt=T\tag{1}$$

Its time average (or mean, or DC value) is given by

$$\bar{s}=\lim_{T_0\rightarrow\infty}\frac{1}{T_0}\int_{-T_0/2}^{T_0/2}s(t)dt=0\tag{2}$$

It is clear that any function for which the integral in (1) is finite, must have a DC-value of zero. The integral in (1) is the value of the Fourier transform of the signal at DC, and this is probably what confuses you. The DC value of a signal, and the value of its Fourier transform at DC are not the same. Any signal with a finite Fourier transform at DC has a DC value of zero, i.e. $\bar{s}=0$. Any signal with a non-zero DC value $\bar{s}\neq 0$ has a Dirac delta impulse component in its Fourier transform at DC.

If you write a signal as

$$s(t)=\bar{s}+\tilde{s}(t)$$

where $\bar{s}$ is the DC component as computed from (2), and, consequently, $\tilde{s}(t)$ has a DC component of zero, then its Fourier transform is

$$S(f)=\bar{s}\delta(f)+\tilde{S}(f)$$

where $\tilde{S}(0)$ is finite.

EDIT: Also note that when the Fourier transform of a signal $s(t)$ has a certain non-zero value at a frequency $f_0$, then this does not entail that the signal has a pure sinusoidal component at that frequency. The same is true for DC. If the Fourier transform has a finite value at DC, the time-domain signal has no DC component, otherwise there would be a Dirac impulse at $f=0$, just as there would be a Dirac impulse at $f_0$ if the signal contained a sinusoid at the frequency.