FT of Signum Function

The Fourier transform of the signum function sgn(t) reveals that an odd, bipolar signal produces a purely imaginary, odd spectrum proportional to 1/(j*pi*f). This result is the building block for finding the Fourier transform of the unit step function and for defining the Hilbert transform.

Core Concept

The signum function is defined as sgn(t) = +1 for t > 0, -1 for t < 0, and 0 for t = 0. It is not absolutely integrable because its magnitude stays at 1 forever. This means the standard Fourier transform integral does not converge directly, and a limiting approach is needed.

The standard trick is to write sgn(t) as the limit of the signal e^(-a|t|)*sgn(t) as a approaches 0. This damped version is absolutely integrable and has a known FT. Taking the limit as a goes to 0 gives the Fourier transform of sgn(t). The result is 1/(j*pi*f), which is purely imaginary and odd.

The signum function is closely linked to the unit step: u(t) = (1 + sgn(t))/2. Its FT is central to the definition of the Hilbert transform, which produces a 90-degree phase shift in all frequency components of a signal. This is used in SSB modulation in radio communications.

Key Formula

FT pair: sgn(t) <-> 1/(j*pi*f) = -j/(pi*f). This can also be written as 2/(j*omega) in the angular frequency convention. The spectrum is purely imaginary for all f not equal to 0, and it is an odd function: X(-f) = -X(f). The magnitude |X(f)| = 1/(pi|f|) grows without bound as f approaches 0, which is consistent with the non-integrable nature of sgn(t).

Derivation using limiting approach:

Step 1: Define approximating signal
  x_a(t) = e^(-at) u(t) - e^(at) u(-t),  a > 0
  Note: x_a(t) -> sgn(t) as a -> 0

Step 2: FT of e^(-at)u(t)
  = 1/(a + j2*pi*f)

Step 3: FT of e^(at)u(-t)
  Note: e^(at)u(-t) = e^(-a(-t))u(-t)
  FT = 1/(a - j2*pi*f)  [by time-reversal]

Step 4: FT of x_a(t)
  X_a(f) = 1/(a + j2*pi*f) - 1/(a - j2*pi*f)
          = [(a - j2*pi*f) - (a + j2*pi*f)] / (a^2 + 4*pi^2*f^2)
          = -j4*pi*f / (a^2 + 4*pi^2*f^2)

Step 5: Take limit as a -> 0
  X(f) = lim_{a->0} [-j4*pi*f / (a^2 + 4*pi^2*f^2)]
        = -j4*pi*f / (4*pi^2*f^2)
        = -j / (pi*f)
        = 1/(j*pi*f)

Final Answer: FT{sgn(t)} = 1/(j*pi*f)

Exam Tip: The FT of sgn(t) is 1/(j*pi*f), not 1/(j*omega). In the omega convention it becomes 2/(j*omega). Confusing these two is a common error. Also remember that sgn(t) itself is not absolutely integrable, so you cannot apply the standard FT formula directly; the limit method or the step-function decomposition must be used.

Properties Summary

• FT pair: sgn(t) <-> 1/(j*pi*f), using the f-convention.
• In omega convention: sgn(t) <-> 2/(j*omega).
• Spectrum is purely imaginary and odd: Re{X(f)} = 0 for all f.
• Relation to step: FT{u(t)} = (1/2)*delta(f) + 1/(j2*pi*f), derived using sgn(t).
• Hilbert transform: defined by H{x(t)} = x(t) * (1/(pi*t)), and the FT of 1/(pi*t) is -j*sgn(f).
• sgn(t) is real and odd, so X(f) is purely imaginary and odd (skew-Hermitian symmetry).

Quick Revision

• sgn(t) = +1 for t > 0, -1 for t < 0.
• u(t) = [1 + sgn(t)] / 2.
• FT{sgn(t)} = 1/(j*pi*f) in f-convention; 2/(j*omega) in omega-convention.
• Derived via limit of e^(-a|t|)*sgn(t) as a -> 0.
• Spectrum is purely imaginary and odd.
• |X(f)| = 1/(pi|f|) diverges at f = 0, consistent with the DC content being undefined.
• Exam trap: writing FT{sgn(t)} = 1/(pi*f) without the j in the denominator. The j is essential: it makes the spectrum purely imaginary and correctly encodes the 90-degree phase relationship between sgn(t) and its spectrum.