# Discrete Fourier Transform: What is the DC Term really?

I am currently toying around with the Discrete Fourier Transform (DFT) in Matlab to extract features from images. I like to fully understand the concepts that I use. I have read several explanations, such as this, but so far, none really explained the meaning of the "DC term". All I know is that the k'the term of the DFT can be written as:

$F_k=\sum_{j=0}^{N-1}{x_j\omega^{k\,j}}$ where $\omega$ is the twiddle factor.

That means that the first term (the DC term), $F_0=\sum_{j=0}^{N-1}{x_j}$, is an amplitude without frequency.

Can someone explain why is it called the DC term? What is it's relation to "Direct Current"? And what is the relevance of the DC term? When is it useful, and for what?

The DC term is the 0 Hz term and is equivalent to the average of all the samples in the window (hence it's always purely real for a real signal). The terminology does indeed come from AC/DC electricity - all the non-zero bins correspond to non-zero frequencies, i.e. "AC components" in an electrical context, whereas the zero bin corresponds to a fixed value, the mean of the signal, or "DC component" in electrical terms.

As far as practical applications go, the DC or 0 Hz term is not particularly useful. In many cases it will be close to zero, as most signal processing applications will tend to filter out any DC component at the analogue level. In cases where you might be interested it can be calculated directly as an average in the usual way, without resorting to a DFT/FFT.

• What is it's use in DSP and maybe statistically speaking?
– Domi
Dec 4 '13 at 4:02
• See edit above - short answer: it's not particularly useful, in my experience at least. Dec 4 '13 at 9:41
• Thank you. I hope you don't mind, I'll leave the question open for a little while longer.
– Domi
Dec 4 '13 at 12:48
• Imagine a sine wave where the zero crossing is at 5 not 0. If you average over the sine wave, all the values cancel out to zero, exception the offset. That's the DC term. In images, it's waaaay less intuitive. The frequencies are the occurances of spatial patterns, think chess board pattern. I'm not sure what the DC term would really be in that case, I think it's the background intensity. Someone else will know...
– wbg
Oct 24 '18 at 6:17

The term "DC" comes from the field of signal processing back when signals were actually small currents on a copper wire... An electrical signal was usually transmited as a small modulation ("AC") over a strong and fixed current/volatage ("DC"). The strong fixed current usually determined the electrical properties of the analog components of the circuit designed to "process" the signal. For example, an amplifier composed of several transistors will have different resistance/delays values depending on the input DC. Thus the designer of the analog circuit will specify a certain DC over which the signal can be "processed" properly by the device.

there are many types of signals. In fact, a numer of signals have a frequency in reality. And dc means not ac. but ac signals have a little or appropriate frequency. In that dc has no frequency but straight signal, dc term has no frequency, F(f=0).

I don't know how it works with images (see below), but with audio you have a waveform that moves into the positive and negative around a center line. This is usually at zero (audio is -1 -> +1), but if there is an offset, ie. if add up all the positive and negative sample values and end up above or below zero, that is the DC offset.

In other words, '0Hz' means this value is applied to the entire signal, ie. it is added (as an offset) to every sample. with audio that is generally undesirable (some algorithms don't like it), so if a particular recording or ADC converter produced an unwanted DC offset you can use a DC filter to remove it.

With images, intuitively I would assume it works the same way. ie. that it's the darkest value in the image. eg. if the darkest pixel isn't true black but (say) dark gray, that gray value would be the DC/offset value (from true black) applied to every other pixel. But I don't know if that's how it actually works with images.