4.4.1 Rectangular Pulse (rect(t/T))

Time Domain Definition:

The rectangular pulse, denoted as rect(t/T) or Π(t/T), is a signal that has a constant amplitude of 1 for a duration of T seconds, centered at t=0, and is 0 everywhere else.

x(t) = 1, for |t| <= T/2
x(t) = 0, for |t| > T/2

Derivation of FT:

X(j*omega) = Integral from t = -T/2 to t = +T/2 of (1 * e^(-j * omega * t) dt)
Solving this integral involves direct integration of the complex exponential.

Fourier Transform Result:

X(jomega) = T * sinc(omega * T / (2pi))
(Note: The sinc function is defined as sinc(x) = sin(pix) / (pix). Sometimes the unnormalized sinc, Sa(x) = sin(x)/x, is used, in which case the argument would be omega*T/2).

Interpretation of Spectrum:

• The spectrum of a rectangular pulse is a sinc function.
• The main lobe of the sinc function is centered at omega = 0 (DC).
• The width of the main lobe (from the first zero crossing on one side to the first zero crossing on the other) is inversely proportional to the pulse duration T. Specifically, the first zero crossing occurs at omega = 2*pi / T.
• Key Insight: A wider pulse in the time domain (larger T) results in a narrower main lobe in the frequency domain, meaning its energy is concentrated in a smaller bandwidth. Conversely, a narrower pulse in time (smaller T) spreads its energy over a wider range of frequencies. This vividly demonstrates the inverse relationship between time duration and bandwidth.

4.4.2 Unit Impulse Function (delta(t))

Time Domain Definition:

The Dirac delta function, delta(t), is an idealized signal (not a regular function) with infinite amplitude at t=0, zero amplitude elsewhere, and an area of 1.

Derivation of FT:

X(jomega) = Integral from -infinity to +infinity of (delta(t) * e^(-j * omega * t) dt)
Using the sifting property of the impulse function (Integral of f(t)delta(t-t0) dt = f(t0)), with t0 = 0 and f(t) = e^(-jomegat), we get:

Fourier Transform Result:

X(j*omega) = e^(j * omega * 0) = 1

Interpretation of Spectrum:

The Fourier Transform of a unit impulse is a constant value of 1 across all frequencies. This means that the impulse function contains all frequencies with equal amplitude and zero phase shift. This is consistent with its time-domain nature: a perfectly sharp, instantaneous event requires an infinite range of frequency components to be accurately represented.

4.4.3 Unit Step Function (u(t))

Time Domain Definition:

The unit step function, u(t), is 0 for t < 0 and 1 for t >= 0.

Derivation of FT (Conceptual/Property-Based):

• The unit step function is not absolutely integrable, so its Fourier Transform is defined using generalized functions (specifically, the impulse function).
• We know that the derivative of the unit step is the unit impulse: d/dt u(t) = delta(t).
• Using the Differentiation in Time property: F{d/dt u(t)} = j*omega * F{u(t)}.
• Since F{delta(t)} = 1, we have: 1 = j*omega * F{u(t)}.
• Solving for F{u(t)} yields F{u(t)} = 1 / (j*omega). However, this misses the DC component.
• A more rigorous approach accounts for the average value (DC component) of u(t), which is 1/2 over infinite time. The full derivation usually involves analyzing u(t) as the limit of a decaying exponential or using properties.

Fourier Transform Result:

X(jomega) = (1 / (jomega)) + pi * delta(omega)

Interpretation of Spectrum:

• The term (1 / (j*omega)) represents the frequency content associated with the step change. It shows that higher frequencies are attenuated (as omega increases, 1/omega decreases).
• The term pi * delta(omega) represents the DC component (zero frequency component). The unit step function has an average value of 1/2 over infinite time, which contributes to the impulse at DC in the frequency domain.

4.4.4 Exponential Signals

Real Exponential (Decaying, e^(-at)u(t) for a > 0):

• Time Domain Definition: x(t) = e^(-at)u(t). This signal starts at t=0 and decays exponentially as t increases.
• Derivation of FT:
  X(jomega) = Integral from t = 0 to t = +infinity of (e^(-at) * e^(-j * omega * t) dt)
  X(jomega) = Integral from t = 0 to t = +infinity of (e^(-(a + j*omega) * t) dt)
  Solving this integral (which is a standard improper integral).
• Fourier Transform Result:
  X(jomega) = 1 / (a + jomega)
• Interpretation of Spectrum: This is a complex spectrum. The magnitude |X(j*omega)| = 1 / sqrt(a^2 + omega^2) shows that the spectrum is highest at omega=0 (DC) and rolls off as frequency increases. This is consistent with a decaying signal that has more low-frequency content.

Complex Exponential (e^(jomega0t)):

• Time Domain Definition: x(t) = e^(jomega0t). This is an infinitely long, pure complex sinusoid.
• Derivation of FT (Conceptual/Property-Based): This signal is not absolutely integrable, so its FT is defined using generalized functions (impulses), usually by using the duality property or by considering it as the inverse FT of a single impulse.
• Fourier Transform Result:
  X(jomega) = 2pi * delta(omega - omega0)
• Interpretation of Spectrum: A pure complex exponential in the time domain corresponds to a single impulse in the frequency domain located exactly at its angular frequency (omega0). This means the signal consists of only one frequency component, as expected.

4.4.5 Sinusoidal Signals (cos(omega0t) and sin(omega0t))

• Using Euler's Formula and Linearity: We derive the FT of real sinusoids by expressing them as a sum of complex exponentials using Euler's formula and then applying the linearity property and the FT of a complex exponential.

Cosine (cos(omega0*t)):

• Time Domain Definition: x(t) = cos(omega0t) = (1/2) * (e^(jomega0t) + e^(-jomega0*t))
• Derivation of FT:
  F{cos(omega0t)} = F{(1/2) * e^(jomega0t)} + F{(1/2) * e^(-jomega0t)}
  (by linearity)
  = (1/2) * [2pi * delta(omega - omega0)] + (1/2) * [2*pi * delta(omega - (-omega0))]
• Fourier Transform Result:
  X(j*omega) = pi * [delta(omega - omega0) + delta(omega + omega0)]
• Interpretation of Spectrum: The spectrum of a cosine wave consists of two impulses, one at positive frequency (+omega0) and one at negative frequency (-omega0), each with a strength of pi. This confirms that a real cosine wave is composed of two complex exponentials rotating in opposite directions.

Sine (sin(omega0*t)):

• Time Domain Definition: x(t) = sin(omega0t) = (1/(2j)) * (e^(jomega0t) - e^(-jomega0*t))
• Derivation of FT: Similar to cosine, applying linearity.
• Fourier Transform Result:
  X(jomega) = jpi * [delta(omega + omega0) - delta(omega - omega0)]
• Interpretation of Spectrum: The spectrum of a sine wave also consists of two impulses at +omega0 and -omega0, but with opposite signs and multiplied by 'j'. This means the components are 90 degrees out of phase compared to the cosine components, which is consistent with the sine and cosine relationship.