6.B Negative Frequencies

Virtually every text book on Fourier analysis treats the introduction of negative frequencies as a natural occurrence, one that is merely a convention, not worthy of any justification.  Yet, in our experience, this concept is one of the least understood basic tenants of Fourier analysis and, consequently, it is often ill-applied by students.  Actually, as shown by the symmetry properties of a real-valued function in the time or space domain (Section 3.3), there is no new information in the negative frequency spectrum. 

A common way to describe the idea of negative frequencies is to visualize a wheel rotating in one direction and then reversing the direction. Rotating in say the counterclockwise (CCW) direction illustrates positive frequency and clockwise (CW) rotation describes negative frequency.  The rotating wheel view is a perfectly correct way of interpreting the + and - frequencies of the complex Fourier spectrum as we will now show.We will justify this statement by providing a detailed understanding of what the Fourier transform of A1 cos (2pf1t) in equation 3.4.1 actually means. Equation 3.4.1 is repeated here as equation B.1.

(B.1)

First, we recall Euler’s relation (equation 3.5b) that decomposes eif into real and imaginary parts, i.e.,

(B.2)

from which we can write,

(B.3)

The inverse of Euler’s relation allows us to express the trigonometric functions as

(B.4)

and

(B.5)

Concentrating on the cosine relationship, the term A1/2 ei2pf1t can be mapped as a vector with real and imaginary parts A1/2 cos (2pf1t) and A1/2 sin (2pf1t), respectively, as in Figure B.1a.  Similarly, A1/2 e-i2pf1t is plotted in Figure B.1b.

Figure B.1b. Mapping of complex exponentials as vectors with real and imaginary parts: A1/2 e-i2pf1t.
Figure B.1a. Mapping of complex exponentials as vectors with real and imaginary parts: A1/2 ei2pf1t

 

 

The key observation is that since f = +2pf1t is an angle that varies linearly with time, the vectors e+i2pf1t also vary with time.  For example, at t = 0 the e+i2pf1t vectors lie along the positive, horizontal axis; at a time t = 1 s later the e+i2pf1t vector has rotated through a CCW angle of 2pf1.  At an arbitrary time t, the rotating e+i2pf1t vector has an angle of 2pf1t CCW from the positive real axis.  The e-i2p f1t vector, with the negative exponent, rotates similarly but in a CW direction.  In Figure B.2a the two counter rotating vectors from Figures B.1a and b are summed as they rotate CCW and CW each with angular velocity w1 = 2pf1 radians/s.  In other words, Figure B.2a graphically performs

(B.6)

 

Figure B.2a. Development of positive and negative frequencies: A1 cos (2pf1t) is sum of two counter rotation vectors from Figure B.1.
Figure B.2b. Development of positive and negative frequencies: A1 sin (2pf1t) is the difference of two counter rotation vectors from Figure B.1.

 

This clearly shows that the cosine function can be viewed as being composed of both positive (CCW) and negative (CW) frequency components. These are expressed in the frequency domain after Fourier transform as d-functions at f = +f1 in equation B.1.  It is also clear from equation B.6 and the vector summation in Figure B.2a that the amplitudes of the d-functions must be A1/2, not A1.  Equation B.5 tells us that the picture for the sine function is that of a difference between counter rotating vectors both of which have imaginary values (Figure B.2b).