8.01 Foundation of the Continuous-Time Fourier Transform CTF

8.00 Chapter Map - Continuous-Time Fourier Transform | 8.02 Properties of the Continuous-Time Fourier Transform

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8.01 Foundation of the Continuous-Time Fourier Transform (CTFT)

Core Idea

The Continuous-Time Fourier Transform (CTFT) is the mathematical extension of the Fourier series to aperiodic (non-periodic) signals of infinite duration [3.20, 255]. By conceptually modeling an aperiodic signal as a periodic signal whose fundamental period approaches infinity ($T_{0}o\infty$), the discrete, harmonically spaced line spectrum collapses into a continuous frequency spectrum [3.46, 256]. This note details the rigorous limiting derivation of the CTFT pair, establishes its existence/convergence criteria, and analyzes the physical impact of time-domain expansion on spectral density.

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1. The Conceptual Transition: Periodic to Aperiodic

For periodic signals, the Fourier series decomposes the signal into discrete frequency components spaced at integer multiples of the fundamental frequency \Omega_0 = rac{2\pi}{T_0} [3.21, 3.22]. However, most real-world transient signals {such as speech, lightning, or a single radar pulse} are aperiodic—they occur once and do not repeat [3.44, 261].

To analyze these signals in the frequency domain, we apply a limiting thought experiment [3.44]:

1. Take an aperiodic signal $x(t)$ of finite duration $2T_1$, such that $x(t)=0$ for $|t|>T_1$ [3.44].
2. Construct a periodic extension $ildex(t)$ by repeating $x(t)$ at regular intervals of $T_0$ (where $T_0>2T_1$) [3.44]: $ildex(t)={\sum }_{k=-\infty }^{\infty }x(t-k{T}_0)$
3. If we now let the repetition period $T_{0}o\infty$, the adjacent pulses are pushed infinitely far away to the left and right [3.46, 256]. The periodic extension $ildex(t)$ converges exactly to our original aperiodic signal $x(t)$ [3.46]: ${\lim }_{T_{0}o\infty }ildex(t)=x(t)$

Aperiodic Signal x(t):
                 _
  ______________| |______________   (Occurs once, zero elsewhere)
               -T1 T1

Periodic Extension x~(t) with Period T0:
         _               _               _
  ______| |_____________| |_____________| |______
        -T0             0               T0
         |<--- T0 ----->|

As the period $T_0$ increases, the fundamental frequency \Omega_0 = rac{2\pi}{T_0} becomes progressively smaller, causing the discrete line spectrum to pack closer together. When $T_{0}o\infty$, the discrete line spacing becomes an infinitesimal step $d\Omega$, turning the discrete sum into a continuous integral [3.46, 256].

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2. Rigorous Derivation of the CTFT Pair

Let us mathematically execute this limiting process.

Step 1: Write the Exponential Fourier Series of the Periodic Extension

The periodic signal $ildex(t)$ with period $T_0$ can be represented using its complex exponential Fourier series [3.21, 3.22]: $ildex(t)={\sum }_{n=-\infty }^{\infty }{C}_n{e}^{jn{\Omega }_{0}t}ext---(Equation1.1)$ where the discrete complex coefficients $C_n$ are calculated over a single period [3.23]: C_n = rac{1}{T_0} \int_{-T_0/2}^{T_0/2} ilde{x}(t) e^{-j n \Omega_0 t} \, dt \quad ext{--- (Equation 1.2)}

Step 2: Relate the Coefficients to the Isolated Pulse Envelope

Since $ildex(t)=x(t)$ in the fundamental interval $[-T_0/2,T_0/2]$ and $x(t)=0$ outside this window, we can expand the limits of integration of Equation 1.2 to infinity [3.45]: C_n = rac{1}{T_0} \int_{-\infty}^{\infty} x(t) e^{-j n \Omega_0 t} \, dt \quad ext{--- (Equation 1.3)}

Let us define a continuous, smooth envelope function of a continuous frequency variable $\Omega$ as [3.45]: $X(j\Omega )={\int }_{-\infty }^{\infty }x(t)e^{-j\Omega t}dtext---(Equation1.4)$ Comparing Equation 1.3 and Equation 1.4 reveals that the discrete coefficients $C_n$ are simply scaled, uniformly sampled points of this continuous spectral envelope $X(j\Omega )$ at discrete harmonic frequencies $\Omega =n{\Omega }_0$ [3.45]: C_n = rac{1}{T_0} X(j n \Omega_0) \quad ext{--- (Equation 1.5)}

Step 3: Substitute the Sampled Envelope back into the Synthesis Equation

Substitute the coefficient expression (Equation 1.5) back into the Fourier series expansion (Equation 1.1) [3.46]: ilde{x}(t) = \sum_{n=-\infty}^{\infty} rac{1}{T_0} X(j n \Omega_0) e^{j n \Omega_0 t} Since \Omega_0 = rac{2\pi}{T_0} \implies rac{1}{T_0} = rac{\Omega_0}{2\pi}, we rewrite the summation as [3.46]: ilde{x}(t) = rac{1}{2\pi} \sum_{n=-\infty}^{\infty} X(j n \Omega_0) e^{j n \Omega_0 t} \Omega_0 \quad ext{--- (Equation 1.6)}

Step 4: Take the Limit as the Period $T_{0}o\infty$

As the period $T_0$ approaches infinity, we observe the following limiting transformations [3.46]:

1. Infinitesimal Frequency Spacing: The fundamental frequency interval ${\Omega }_0$ approaches zero and is represented by the differential frequency element $d\Omega$ [3.46]: ${\Omega }_{0}od\Omega$
2. Continuous Frequency Variable: The discrete harmonic frequencies $n{\Omega }_0$ merge into a continuous frequency variable $\Omega$: $n{\Omega }_{0}o\Omega$
3. Summation to Integration: The infinite sum over discrete intervals of width ${\Omega }_0$ becomes a continuous definite integral over all real frequencies [3.46]: ${\sum }_{n=-\infty }^{\infty }[\cdot ]{\Omega }_{0}o{\int }_{-\infty }^{\infty }[\cdot ]d\Omega$
4. Signal Convergence: The periodic extension $ildex(t)$ converges back to the original unique transient signal $x(t)$ [3.46]: $ildex(t)ox(t)$

Applying these limit conditions directly to Equation 1.6 yields the Inverse Continuous-Time Fourier Transform (IDFT / Synthesis Equation) [3.46]:

\mathbf{x(t) = rac{1}{2\pi} \int_{-\infty}^{\infty} X(j\Omega) e^{j\Omega t} \, d\Omega} \quad ext{--- (Equation 1.7)}

where the spectral envelope $X(j\Omega )$ is defined by the Forward Continuous-Time Fourier Transform (CTFT / Analysis Equation) [3.46]:

$\mathbf{X}(\mathbf{j\Omega })={\int }_{-\infty }^{\infty }\mathbf{x}(\mathbf{t}){\mathbf{e}}^{-\mathbf{j\Omega t}}\mathbf{dt}ext---(Equation1.8)$

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3. The Continuous-Time Fourier Transform Pair

Equations 1.7 and 1.8 form the Fourier Transform Pair [3.46, 320]. We denote this mutual mathematical linkage as: $x(t)⟷X(j\Omega )$ or using operator notation: $X(j\Omega )=\mathcal{F}\{x(t)\}extandx(t)={\mathcal{F}}^{-1}\{X(j\Omega )\}$

                FORWARD CTFT: F{ x(t) }
   +-------------------------------------------------+
   |                                                 |
   v                                                 v
 [Time Domain]                                 [Frequency Domain]
  x(t) (Volts)                                  X(jΩ) (Volts·sec)
   ^                                                 ^
   |                                                 |
   +-------------------------------------------------+
               INVERSE CTFT: F^-1{ X(jΩ) }

Physical Interpretation of Units and Dimensions

• Time-Domain Signal $x(t)$: Typically measures amplitude over time {e.g., Volts ($V$) or Amperes ($A$)}.
• Fourier Spectrum $X(j\Omega )$: Represents spectral density rather than individual discrete amplitudes. Integrating Equation 1.8 shows that the units of $X(j\Omega )$ are amplitude multiplied by time {e.g., Volt-seconds ($V\cdot exts$) or Volts per Radian/sec ($V/(extrad/s)$)}.
• Therefore, $X(j\Omega )$ does not show the amplitude of a single discrete frequency (which is infinitesimally small for an aperiodic signal); it shows the continuous density of frequencies across the spectrum.

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4. Existence and Convergence: The Dirichlet Conditions

Because the limits of the continuous Fourier integrals extend to infinity ($-\infty$ to $\infty$), we must establish strict conditions to guarantee that the integrals converge mathematically.

The Dirichlet Conditions for the CTFT

For an aperiodic signal $x(t)$ to have a valid, well-defined Fourier transform $X(j\Omega )$, it must satisfy the following three conditions [3.11, 261]:

1. Absolute Integrability: The signal must be absolutely integrable over the entire real time-line [3.11]: ${\int }_{-\infty }^{\infty }|x(t)|dt<\infty$ Proof of bounding: Taking the absolute magnitude of Equation 1.8 yields:

ight| \le \int_{-\infty}^{\infty} \left| x(t) e^{-j\Omega t} ight| , dt = \int_{-\infty}^{\infty} |x(t)| , dt$$

Since the integral is bounded by a finite value, the spectrum $X(j\Omega)$ is guaranteed to be finite and free from infinite spikes [3.11].

2. Finite Extrema: $x(t)$ must have a finite number of local maxima and minima within any finite interval of time.

3. Finite Discontinuities: $x(t)$ must have a finite number of discontinuities within any finite time interval, and each discontinuity must have a bounded, finite height.

4.1 Bounded Convergence and the Gibbs Midpoint Rule

If $x(t)$ satisfies the Dirichlet conditions and has a jump discontinuity at $t=t_0$, the inverse Fourier transform integral does not diverge [3.11]. Instead, it converges exactly to the arithmetic midpoint of the discontinuity: \mathcal{F}^{-1}\{X(j\Omega)\} \Big|_{t=t_0} = rac{x(t_0^+) + x(t_0^-)}{2}

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5. High-Yield Exam Analytics: Changing the Period $T_0$

High-Yield 5-Mark Exam Classic [PYQ 2016]

What is the Fourier transform? What are the effects on the discrete spectrum of a periodic signal if its time period ($T_0$) increases?

Step-by-Step Analytical Explanation Plan:

1. Define the Fourier Transform: State that the CTFT converts a continuous time-domain signal $x(t)$ into a continuous frequency-domain representation $X(j\Omega )$ [3.46]. Write Equations 1.7 and 1.8.
2. The Discrete Harmonic Spacing Formula: Show that the periodic signal has discrete spectral components located at: \Omega_n = n\Omega_0 = n rac{2\pi}{T_0}
3. Effect 1: Spectral Line Density Compression: As the time period $T_0$ increases, the fundamental frequency spacing ${\Omega }_0=2\pi /T_0$ decreases. The discrete spectral lines become denser and pack more tightly together.
4. Effect 2: Amplitude Scaling Attenuation: The amplitude of the discrete Fourier series coefficients C_n = rac{1}{T_0} X(j n \Omega_0) scales down inversely with $T_0$ [3.45]. As $T_{0}o\infty$, the discrete coefficients $C_{n}o0$.
5. Effect 3: Continuous Spectral Transition: Despite the amplitudes shrinking to zero, the relative ratio of the coefficients is locked to the shape of the continuous pulse envelope $X(j\Omega )$ [3.45]. In the limit as $T_{0}o\infty$, the discrete line spectrum transitions completely into a smooth continuous frequency spectrum [3.46, 256].

  T0 is small (Discrete, sparse spacing):
      |       |       |       |       |       |       |
    --o-------o-------o-------o-------o-------o-------o--
             -3Ω0    -2Ω0    -Ω0      0      Ω0      2Ω0     3Ω0

  T0 is large (Discrete, dense spacing):
    |||||||||||||||||||||||||||||||||||||||||||||||||||||
    --o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o--

  T0 -> infinity (Continuous spectrum):
    _____________________________________________________
    ----------------------------------------------------- (Continuous line)

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6. Common Mistakes That Cost Marks

The Cyclic Frequency vs. Angular Frequency Scaling Trap

ECE 2107 utilizes angular frequency $\Omega$ (in rad/s), while some secondary mathematics textbooks (such as Math 2109) use cyclic frequency $f$ (in Hz).

• If you write the Fourier Transform pair using angular frequency $\Omega$, you must include the scaling factor of rac{1}{2\pi} in the Inverse Transform [3.46]: x(t) = rac{1}{2\pi} \int_{-\infty}^{\infty} X(j\Omega) e^{j\Omega t} \, d\Omega
• If you write the pair using cyclic frequency $f$ (where $\Omega =2\pi f$), the scaling factor disappears: $x(t)={\int }_{-\infty }^{\infty }X(f)e^{j2\pi ft}dfextandX(f)={\int }_{-\infty }^{\infty }x(t)e^{-j2\pi ft}dt$
• Exam Danger: Mixing these two notations {e.g., omitting the rac{1}{2\pi} factor while integrating with respect to $d\Omega$} is a critical error that results in a zero-mark grading on derivations.

The Non-Convergent Integration Trap

Attempting to evaluate the CTFT of non-absolutely integrable signals {such as a constant $x(t)=1$ or a unit step $u(t)$} using the standard integral formula (Equation 1.8) results in an invalid, divergent calculation because ${\int }_{-\infty }^{\infty }1dt=\infty$.

• To find the CTFT of these signals, you must use distribution-domain modeling with Dirac delta functions (which we cover in Note 8.04).

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7. PYQ Bank — Verbatim Questions & Answer Plans

7.1 PYQ 2015 [5 Marks]

Question: Define the Fourier transform of a time function and explain under what condition it exists.

• Answer Plan:
  1. Define the Fourier Transform mathematically as an integral operation that converts a time function $x(t)$ to its continuous frequency spectrum $X(j\Omega )$ [3.20, 320].
  2. Write the Forward CTFT equation (Equation 1.8).
  3. State the three Dirichlet Conditions (Absolute Integrability, Bounded Extrema, Bounded Discontinuities) as detailed in Section 4 [3.11, 261]. Highlight that absolute integrability guarantees $|X(j\Omega )|$ is bounded [3.11].

7.2 PYQ 2025/2024 [10 Marks]

Question: Define Fourier transform. Briefly explain the properties of Fourier transform.

• Answer Plan:
  1. Provide the formal definition and write down the complete CTFT / Inverse CTFT equation pair [3.46].
  2. Define a Fourier Transform Pair as a specific mathematical relationship linking a unique time-domain function $x(t)$ with its corresponding continuous frequency-domain representation $X(j\Omega )$.
  3. Summarize the major physical properties (Linearity, Time Shifting, Time Scaling, Modulation, and Convolution) using a concise properties table. (Refer to Note 8.02 for the complete property proofs).

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8. Self-Check Before Moving On

• Can you derive the Inverse Fourier Transform integral by taking the limit of the Fourier series as $T_{0}o\infty$? [2.0]
• Do you know where the rac{1}{2\pi} scaling factor belongs in the CTFT equations, and why it is there? [6.0]
• Can you state the three Dirichlet convergence conditions for aperiodic signals? [4.0]
• Do you understand why the continuous Fourier transform measures frequency density instead of discrete amplitude spikes? [3.0]

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Source: (k.Deergha Rao) signals and systems.pdf (Chapter 3), 02 Fourier Transform.pdf, lec 3 u academy online playlist.pdf, Rabiul sir class note.pdf.