8.00 Continuous-Time Fourier Transform CTFT Compact Review

Chapter 8: Continuous-Time Fourier Transform (CTFT) - Compact Review

7.00 Continuous-Time Fourier Series - Compact Review | 9.00 Laplace Transform & s-Domain Circuit Applications - Compact Review

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8.01 Foundation of the CTFT & Existence Conditions

*(Target: Theory Descriptive / Mathematical Proof / 10-Mark Derivation)*

• Concept: Represents aperiodic, transient signals in the continuous frequency domain by modeling them as periodic signals whose fundamental period approaches infinity ($T_{0}o\infty$).
• The CTFT limiting derivation steps:
  1. Express periodic counterpart $ildex(t)$ with fundamental period $T_0$ and spacing \omega_0 = rac{2\pi}{T_0} using the complex exponential Fourier series: ilde{x}(t) = \sum_{n=-\infty}^{\infty} C_n e^{j n \omega_0 t} \quad ext{where} \quad 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{[221, 236, 240]}
  2. Substitute $C_n$ back into the series, scale terms with rac{\omega_0}{2\pi} = rac{1}{T_0}, and set the limit as $T_{0}o\infty$ (where fundamental frequency spacing ${\omega }_{0}od\omega$ and discrete index $n{\omega }_{0}o\omega$ becomes a continuous variable) (derivation):
• Forward Fourier Transform: $F(j\omega )={\int }_{-\infty }^{\infty }f(t)e^{-j\omega t}dtext[124,225,240,344]$
• Inverse Fourier Transform: f(t) = rac{1}{2\pi} \int_{-\infty}^{\infty} F(j\omega) e^{j\omega t} \, d\omega \quad ext{[124, 225, 240, 345]}
• Dirichlet Convergence Conditions:
  • Condition 1 (Absolute Integrability): ${\int }_{-\infty }^{\infty }|f(t)|dt<\infty$ {prevents infinite growth}.
  • Condition 2 (Finite Extremas): Finite number of maxima and minima within any finite interval.
  • Condition 3 (Finite Discontinuities): Finite number of discontinuities within any finite interval, with all jump heights bounded.
• Gibbs Phenomenon in CTFT: Truncating high frequencies causes a permanent 9% overshoot at sharp discontinuities when reconstructing $f(t)$ via a finite frequency band.

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8.02 Master CTFT Transform Pairs

*(Target: Numerical Solving / Rapid Lookups)*

Signal $f(t)$	Fourier Transform $F(j\omega )$	Notes / Physical Interpretation
Unit Impulse $\delta (t)$	$1$	Infinite, flat spectrum; contributes equally to all frequencies.
DC Constant $1$	$2\pi \delta (\omega )$	Spectral impulse located exclusively at DC frequency ($\omega =0$).
Complex Exponential $e^{j{\omega }_{0}t}$	$2\pi \delta (\omega -{\omega }_0)$	Pure phase phasor; a single positive spectral line.
Cosine Wave $\cos ({\omega }_{0}t)$	$\pi \left[ \delta(\omega - \omega_0) + \delta(\omega + \omega_0)
ight]$	Two real, symmetrical line impulses at $\pm {\omega }_0$.
Sine Wave $\sin ({\omega }_{0}t)$	$j\pi \left[ \delta(\omega + \omega_0) - \delta(\omega - \omega_0)
ight]$	Symmetrical imaginary impulses pointing in opposite directions.
Signum Function $extsgn(t)$	rac{2}{j\omega}	Model for a polarity flip; pure imaginary, odd spectrum.
Unit Step $u(t)$	rac{1}{j\omega} + \pi \delta(\omega)	Integrates $\delta (t)$; includes DC impulse ($\pi \delta (\omega )$).
Causal Exponential $e^{-at}u(t)$	rac{1}{a + j\omega} \quad (\Re e\{a\} > 0)	Single-sided decaying transient.
Double Exponential e^{-a\lvert t ert}	rac{2a}{a^2 + \omega^2} \quad (\Re e\{a\} > 0)	Symmetrical infinite-duration decay; yields real Lorentian curve.
Causal Ramp Transient $t{e}^{-at}u(t)$	rac{1}{(a + j\omega)^2} \quad (\Re e\{a\} > 0)	Evaluated using s-domain/frequency differentiation.
Rectangular Gate Pulse $ ext{rect}\left(rac{t}{ au}
ight)$	$ au ext{sa}\left(rac{\omega au}{2}
ight) = au ext{sinc}\left(rac{\omega au}{2\pi}
ight)$	Flat pulse of width $au$ centered at $t=0$; sinc spectrum.
Ideal Low-Pass Filter $rac{W}{\pi} ext{sinc}\left(rac{Wt}{\pi}
ight)$	$ ext{rect}\left(rac{\omega}{2W}
ight)$	Infinitely long sinc impulse response; brick-wall spectrum of width $2W$.
Normalized Gaussian Pulse $e^{-\pi t^2}$	e^{-\pi f^2} = e^{-rac{\omega^2}{4\pi}}	Symmetrical bell curve; transforms exactly into itself.
Delayed Impulse $\delta (t-t_0)$	$e^{-j\omega t_0}$	Pure linear phase delay; constant unit magnitude.

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8.03 Mathematical Properties of the CTFT

*(Target: Theory Descriptive / Mathematical Proof / Numerical Solving)*

• Linearity:

ight} = a_1 X_1(j\omega) + a_2 X_2(j\omega) \quad ext{[124, 126, 226]}$$

• Time Shifting:

ight} = X(j\omega) e^{-j\omega t_0} \quad ext{[124, 126, 226, 346]}$$ Physical Interpretation: Delaying a signal in time shifts its phase linearly with frequency, keeping the magnitude spectrum unchanged.

• Frequency Shifting (Modulation):

ight} = X(j(\omega - \omega_0)) \quad ext{[124, 126, 226, 347]}$$

• Time Scaling:

ight} = rac{1}{\lvert a ert} X\left(jrac{\omega}{a} ight) \quad ext{[124, 126, 226, 348]}$$ Physical Interpretation: Compressing a signal in the time domain ($a>1$) expands its frequency spectrum, showing the reciprocal relationship of time-bandwidth limits.

• Duality:

ight} = X(j\omega) \implies \mathcal{F}\left{ X(jt) ight} = 2\pi x(-\omega) \quad ext{[124, 126, 226, 336]}$$

• Time Differentiation:

ight} = (j\omega)^n X(j\omega) \quad ext{[124, 126, 226, 348]}$$

• Frequency Differentiation:

ight} = rac{dX(j\omega)}{d\omega} \implies \mathcal{F}\left{ t \cdot x(t) ight} = j rac{dX(j\omega)}{d\omega} \quad ext{[124, 126, 226, 333]}$$

• Time Integration:

ight} = rac{X(j\omega)}{j\omega} + \pi X(0) \delta(\omega) \quad ext{[124, 126, 226]}$$

• Time Convolution Theorem:

ight} = X_1(j\omega) X_2(j\omega) \quad ext{[124, 126, 226, 349]}$$

• Frequency Convolution (Windowing/Modulation Theorem):

ight} = rac{1}{2\pi} \left[ X_1(j\omega) * X_2(j\omega) ight] \quad ext{[124, 126, 226]}$$

• Conjugation:

ight} = X^*(-j\omega) \quad ext{[124, 126, 226]}$$

8.03.1 Area Properties

*(Target: Mathematical Proof / 3-Mark Identity)*

• Area Under the Time Curve: ${\int }_{-\infty }^{\infty }f(t)dt=F(0)ext(derivation)ext[124,336]$
• Area Under the Spectral Curve: f(0) = rac{1}{2\pi} \int_{-\infty}^{\infty} F(j\omega) \, d\omega \quad ext{(derivation)} \quad ext{[124, 336]}

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8.04 Waveform Symmetry Conditions & Spectra

*(Target: Theory Descriptive / Symmetry Shortcuts)*

If the time signal $x(t)$ is real, then the CTFT is conjugate symmetric: $X(-j\omega )=X^{\ast }(j\omega )$. Real waveforms exhibit an even magnitude spectrum $|X(j\omega )|=|X(-j\omega )|$ and an odd phase spectrum ngle X(j\omega) = -ngle X(-j\omega).

Table 8.1: CTFT Symmetry Condition Shortcuts

Waveform Condition of $x(t)$	Real Part $X_R(j\omega )$	Imaginary Part $X_I(j\omega )$	CTFT Spectrum $X(j\omega )$
Real & Even	Even	Exactly Zero	Purely Real & Even ($X(j\omega )=X^{\ast }(j\omega )$)
Real & Odd	Exactly Zero	Odd	Purely Imaginary & Odd ($X(j\omega )=-X^{\ast }(j\omega )$)
Imaginary & Even	Exactly Zero	Even	Purely Imaginary & Even
Imaginary & Odd	Odd	Exactly Zero	Purely Real & Odd

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8.05 Rayleigh’s Energy Theorem & Spectral Densities

*(Target: Theory Descriptive / Mathematical Proof)*

8.05.1 Rayleigh’s Energy Theorem

*(Target: Mathematical Proof / 5-Mark Derivation)*

• Theorem Statement: The total energy of an aperiodic signal is identical whether calculated in the time domain or integrated across its energy spectral density in the frequency domain.
• Governing Formula: E_x = \int_{-\infty}^{\infty} \lvert x(t) ert^2 \, dt = rac{1}{2\pi} \int_{-\infty}^{\infty} \lvert X(j\omega) ert^2 \, d\omega \quad ext{(derivation)} \quad ext{[124, 126, 227, 345]}

Table 8.2: Energy Spectral Density (ESD) vs. Power Spectral Density (PSD)

Feature	Energy Spectral Density (ESD) $\Psi (j\omega )$	Power Spectral Density (PSD) $S(j\omega )$
Prerequisite	Energy Signals ($0<E<\infty$, $P=0$)	Power Signals ($E=\infty$, $0<P<\infty$)
Definition	Squared magnitude spectrum: \Psi(j\omega) = \lvert X(j\omega) ert^2	Limit of time-averaged squared magnitude: S(j\omega) = \lim_{T o \infty} rac{\lvert X_T(j\omega) ert^2}{T}
Dimensions	$ext{Volts}^2\cdot ext{s}^2$ or $extJoules/Hz$	$ext{Volts}^2$ or $extWatts/Hz$
Autocorrelation	FT of Energy Autocorrelation: $\mathcal{F}\{R_{xx}(au)\}=\Psi (j\omega )$	FT of Power Autocorrelation: $\mathcal{F}\{R_{xx}(au)\}=S(j\omega )$ (Wiener-Khinchin Theorem)
Total Area	rac{1}{2\pi}\int_{-\infty}^{\infty} \Psi(j\omega) \, d\omega = ext{Total Energy } E	rac{1}{2\pi}\int_{-\infty}^{\infty} S(j\omega) \, d\omega = ext{Total Average Power } P

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8.06 LTI System Analysis & Continuous Networks

*(Target: Circuit Analysis / Numerical Solving)*

Continuous-time systems relate input $x(t)$ to output $y(t)$ using convolution in time, which corresponds to multiplication in the frequency domain: $y(t)=x(t)\ast h(t)\leftrightarrow Y(j\omega )=X(j\omega )H(j\omega )ext[124,349]$

8.06.1 First-Order Low-Pass RC Network Analysis

*(Target: Circuit Analysis / 10-Mark Design Problem)*

              R
        o----/\/\/\-------+--------o Output y(t)
        +                 |        +
     Input x(t)          === C   across Capacitor
        -                 |        -
        o-----------------+--------o

• Network transfer function derivation: Using s-domain/frequency-domain impedances: $Z_R=R$, Z_C = rac{1}{j\omega C}. By voltage divider: H(j\omega) = rac{Y(j\omega)}{X(j\omega)} = rac{rac{1}{j\omega C}}{R + rac{1}{j\omega C}} = rac{1}{1 + j\omega RC} \quad ext{(derivation)} \quad ext{[124, 229]}
• System transient response derivation: For a decaying exponential input $x(t)=e^{-t/RC}u(t)$, find output $y(t)$.
  1. Continuous input spectrum: X(j\omega) = rac{1}{1/RC + j\omega} = rac{RC}{1 + j\omega RC}.
  2. Output spectrum:

ight) \left( rac{1}{1 + j\omega RC} ight) = rac{RC}{(1 + j\omega RC)^2} = rac{1/RC}{(1/RC + j\omega)^2} \quad ext{[124]} 3. Applying the pair $t e^{-at}u(t) \leftrightarrow rac{1}{(a + j\omega)^2}$ where $a = 1/RC$ `(derivation)`: y(t) = rac{t}{RC} e^{-t/RC} u(t) \quad ext{[124, 229]}$$

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8.07 Derivations of High-Yield Core Results

*(Target: Mathematical Proof)*

8.07.1 Normalized Gaussian Pulse is its Own Fourier Transform

*(Target: Mathematical Proof / 5-Mark Derivation)*

• Starting Hypothesis: Let $f(t)=e^{-\pi t^2}$.
• Derivation steps:
  1. Write the forward CTFT integral: $F(j\omega )={\int }_{-\infty }^{\infty }{e}^{-\pi t^2}{e}^{-j\omega t}dtext[124]$
  2. Complete the square in the exponent: -\left( \pi t^2 + j\omega t ight) = -\pi \left( t + rac{j\omega}{2\pi} ight)^2 - rac{\omega^2}{4\pi}.
  3. Pull out the constant term and apply the substitution u = \sqrt{\pi}\left( t + rac{j\omega}{2\pi} ight), knowing the standard Gaussian integral is ${\int }_{-\infty }^{\infty }{e}^{-u^2}du=\sqrt{\pi }$ (derivation):
• Result: F(j\omega) = e^{-rac{\omega^2}{4\pi}} = e^{-\pi f^2} \quad ext{[124, 228, 350]}

8.07.2 Fourier Transform of the Signum Function

*(Target: Mathematical Proof / 5-Mark Derivation)*

• Starting Hypothesis: The signum function is represented as the limit of damped exponentials:

ight] \quad ext{[124, 228]}$$

• Derivation steps:
  1. Apply the forward CTFT integral with damping term $a>0$:

ight] \quad ext{[124]}$$ 2. Evaluate individual integrals: rac{1}{a + j\omega} - rac{1}{a - j\omega} = rac{-2j\omega}{a^2 + \omega^2}. 3. Take the limit as $ao0$ (derivation):

• Result: F(j\omega) = rac{2}{j\omega} = -jrac{2}{\omega} \quad ext{[124, 228, 351]}

8.07.3 Time Shifting Property Proof

*(Target: Mathematical Proof / 5-Mark Derivation)*

• Starting Hypothesis: Show that delaying a signal in time shifts its phase in frequency.
• Derivation steps:
  1. Write the forward CTFT of a shifted signal: $\mathcal{F}\{f(t-t_0)\}={\int }_{-\infty }^{\infty }f(t-t_0)e^{-j\omega t}dtext[124]$
  2. Apply the substitution variable $au=t-t_0⟹t=au+t_0$ and $dt=dau$: $\mathcal{F}\{f(t-t_0)\}={\int }_{-\infty }^{\infty }f(au)e^{-j\omega (au+t_0)}dau=e^{-j\omega t_0}{\int }_{-\infty }^{\infty }f(au)e^{-j\omega au}dauext[124]$
  3. Identify the remaining integral as $F(j\omega )$ (derivation):
• Result: $\mathcal{F}\{f(t-t_0)\}=F(j\omega )e^{-j\omega t_0}ext[124,226,346]$

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8.08 Common Mistakes That Cost Marks

Exam Pitfalls & Marks-Losing Traps

• The Unit Step DC Fallacy: Accidentally evaluating the Fourier transform of the unit step $u(t)$ as simply rac{1}{j\omega}. Since $u(t)$ has a non-zero average DC value, you must include the impulse component: rac{1}{j\omega} + \pi \delta(\omega). Failing to write $\pi \delta (\omega )$ will cost you 2 to 3 marks.
• The $1/2\pi$ Inverse Scaling Omission: Omitting the scaling constant rac{1}{2\pi} when evaluating the continuous inverse Fourier transform integral: f(t) = \mathbf{rac{1}{2\pi}} \int F(j\omega) e^{j\omega t} d\omega. (This scaling is not present in Laplace, making it a very common memory leak under pressure).
• Frequency-Scaling vs. Modulation Signs: Confusing the signs of frequency scaling and modulation. Time delay is a negative phase shift ($f(t-t_0)\leftrightarrow F(j\omega )e^{-j\omega t_0}$), whereas time advance is a positive shift. Similarly, modulation by a positive phasor shifts the spectrum to the right ($e^{j{\omega }_{0}t}x(t)\leftrightarrow X(j(\omega -{\omega }_0))$).

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8.09 PYQ Bank — Verbatim Questions & Answer Plans

Q1: The Gaussian Pulse Self-Transform [5-Mark, KUET 2024/2018/2017]

• Question: Show that the normalized Gaussian pulse is its own Fourier transform.
• Answer Plan:
  1. Define the Gaussian pulse as $f(t)=e^{-\pi t^2}$.
  2. Set up the forward CTFT integral and complete the square in the exponent.
  3. Apply the substitution $u=\sqrt{\pi }(t+j\omega /2\pi )$, evaluate the standard integral $\int e^{-u^2}du=\sqrt{\pi }$ (derivation), and write the final result: $F(j\omega )=e^{-{\omega }^2/4\pi }$.

Q2: Time-Domain Convolution to Spectral Multiplication [4/15-Mark, KUET 2022/2018]

• Question: Show that the convolution of signals in the time domain is equal to the multiplication of their individual Fourier transforms in the frequency domain.
• Answer Plan:
  1. Define the convolution integral $y(t)={\int }_{-\infty }^{\infty }{x}_1(au)x_2(t-au)dau$.
  2. Take the forward CTFT of $y(t)$ and change the order of integrations.
  3. Apply the time shifting property on $x_2(t-au)$ inside the integral to isolate $X_2(j\omega )$ (derivation), proving $Y(j\omega )=X_1(j\omega )X_2(j\omega )$.

Q3: Continuous-Time Piecewise Step Pulse [3/15-Mark, KUET 2025/2016/2015]

• Question: Find the Fourier transform of the time function $f(t)=5[u(t+3)+u(t+2)-u(t-2)-u(t-3)]$.
• Answer Plan:
  1. Rewrite $f(t)$ as a sum of two rectangular gate pulses: $f(t)=5extrect(t/6)+5extrect(t/4)$.
  2. Convert each rectangular pulse using the standard pair $extrect(t/au)\leftrightarrow auextsa(\omega au/2)$.
  3. Sum the transforms using the linearity property: F(j\omega) = 30 ext{sa}(3\omega) + 20 ext{sa}(2\omega) = 30rac{\sin(3\omega)}{3\omega} + 20rac{\sin(2\omega)}{2\omega}.

Q4: Linear Phase Shift of Time Shifts [13-Mark, KUET 2023]

• Question: Show that the time shift in the time domain is equal to a phase shift in the frequency domain.
• Answer Plan: See the complete derivation in Section 8.07.3.

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8.10 Self-Check Before Moving On

• Can you write out the forward and inverse CTFT integrals from memory, including the correct location of the $1/2\pi$ scaling constant?
• Do you know all 3 Dirichlet conditions required for CTFT existence?
• Can you prove why odd real signals must have a purely imaginary and odd frequency spectrum?
• Do you understand the difference between ESD and PSD, particularly which one applies to a periodic power signal versus a transient energy pulse?
• Can you derive the transfer function of a first-order passive RC network and calculate its transient output $y(t)$ using the differentiation/scaling property?

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Source: signals and systems (k.Deergha Rao).pdf, continuous Fourier transform properties (02 Fourier Transform.pdf), and class lecture notes (Rabiul sir class note.pdf).