# Interpreting Time Domain Derivative in Frequency Domain

One of the properties of Fourier Transform is that the derivative of a signal in time domain gets translated to multiplication of the signal spectrum by in frequency domain. This property is usually derived as follows.

For a signal with Fourier Transform

we have

which is the inverse Fourier Transform of .

Now we want to understand this relation one level deeper, i.e., the reason behind the factor ? There are two parts of this expression: one is and the other is . We start with .

Notice from the definition of Fourier Transform that this operation decomposes a signal into a sequence of complex sinusoids with frequencies ranging from to . This is shown in Figure 1 below.

Figure 1: Three complex sinusoids and their decomposition into sines and cosines

By Euler’s formula,

Naturally, the higher the frequency, the steeper the slope and hence larger the derivative. After all, a derivative is nothing but the slope of the line tangent to the curve at a point. This is where the factor comes from (simply put, the derivative of is ).

The term is more interesting. The derivative of is while that of is . So from Euler’s formula and using ,

Remembering that , the factor is therefore necessary to rotate and by their corresponding angles such that we get our basis signals back. This results in getting the same signal at the output with multiplication by .