4. Fourier Transform Properties

Fourier Transform (FT) is a powerful tool to analyze signals in the frequency domain. It has many useful properties that make it easier to work with signals. Understanding these properties deeply helps to simplify complex problems and sometimes discover better approaches.

4.1 Linearity

Statement:

The Fourier Transform is a linear operator. This means if you have two signals f(t) and g(t), and constants a and b, then:

Fourier Transform of a linear combination:

F{a f(t) + b g(t)} = a F{f(t)} + b F{g(t)}

Where F{·} denotes the Fourier transform.

Explanation:

• This means the Fourier Transform of a weighted sum of signals equals the weighted sum of their Fourier Transforms.
• This helps to break down complex signals into simpler parts, transform them separately, and then add the results.

Example:

Let

f(t) = cos(2π t), g(t) = sin(2π t), a=2, b=3

Then,

F{2 cos(2π t) + 3 sin(2π t)} = 2 F{cos(2π t)} + 3 F{sin(2π t)}

Using known transforms:

• F{cos(2π t)} = π [δ(ω - 2π) + δ(ω + 2π)]
• F{sin(2π t)} = j π [δ(ω + 2π) - δ(ω - 2π)]

Therefore,

= 2 π [δ(ω - 2π) + δ(ω + 2π)] + 3 j π [δ(ω + 2π) - δ(ω - 2π)]

4.2 Time Shifting

Statement:

If f(t) has Fourier Transform F(ω), then shifting f(t) in time by t₀ results in:

F{f(t - t₀)} = e-j ω t₀ F(ω)

Explanation:

• Shifting a signal in time multiplies its Fourier Transform by a complex exponential factor.
• This factor changes the phase but not the magnitude of the Fourier Transform.
• Important for understanding delays and time shifts in signals.

Example:

Suppose

f(t) = e-t²

Its Fourier Transform is known:

F(ω) = √π e-ω²/4

Now shift by t₀ = 3:

F{f(t - 3)} = e-j ω · 3 √π e-ω²/4 = √π e-ω²/4 e-j 3 ω

• The magnitude |F(ω)| is the same.
• Phase shifted by -3 ω.

4.3 Frequency Shifting

Statement:

If f(t) has Fourier Transform F(ω), then multiplying f(t) by a complex exponential shifts its Fourier Transform in frequency:

F{f(t) ej ω₀ t} = F(ω - ω₀)

Explanation:

• Multiplying a signal by ej ω₀ t shifts the frequency spectrum by ω₀.
• Used extensively in modulation (e.g., radio transmission).

Example:

Given f(t) with Fourier Transform F(ω), multiply by ej 5 t:

F{f(t) ej 5 t} = F(ω - 5)

The entire frequency spectrum moves to the right by 5 units.

4.4 Scaling (Time and Frequency)

Statement:

If you scale the time variable t by a factor a ≠ 0, then:

F{f(a t)} = (1 / |a|) F(ω / a)

Explanation:

• Compressing the signal in time (i.e., |a| > 1) stretches its spectrum in frequency.
• Expanding the signal in time (i.e., |a| < 1) compresses the spectrum.
• The amplitude is scaled by 1 / |a| to conserve energy.

Example:

If f(t) = e-t² with

F(ω) = √π e-ω²/4

Calculate the transform of f(2t):

F{f(2t)} = (1/2) √π e-ω²/16

• The frequency spectrum is stretched by factor 2 (spread out).
• The amplitude is halved.

4.5 Convolution Theorem

Statement:

The Fourier Transform of the convolution of two signals equals the product of their Fourier Transforms:

F{f(t) * g(t)} = F(ω) · G(ω)

Where convolution is defined as:

(f * g)(t) = ∫-∞∞ f(τ) g(t - τ) dτ

Explanation:

• Convolution in time domain corresponds to multiplication in frequency domain.
• This simplifies many signal processing tasks, especially filtering.

Example:

Suppose

f(t) = e-t², g(t) = rect(t)

rect(t) = 1 for |t| < 1/2, else 0.

Their Fourier Transforms:

F(ω) = √π e-ω²/4, G(ω) = sinc(ω/2)

Then:

F{f * g} = F(ω) G(ω) = √π e-ω²/4 · sinc(ω/2)

4.6 Differentiation and Integration in Time Domain

Differentiation:

F{d/dt f(t)} = j ω F(ω)

Explanation:

• Differentiating a signal in time corresponds to multiplying its Fourier Transform by j ω.
• Useful for analyzing systems involving derivatives.

Example:

If

f(t) = e-t², F(ω) = √π e-ω²/4

Then,

F{d/dt e-t²} = j ω √π e-ω²/4

Integration:

F{∫-∞t f(τ) dτ} = (1 / j ω) F(ω) + π F(0) δ(ω)

Explanation:

• Integration in time corresponds to dividing Fourier Transform by j ω, plus a term at zero frequency.
• Used in system analysis and control.

4.7 Parseval’s Theorem

Statement:

Parseval’s theorem relates total energy in time domain to energy in frequency domain:

∫-∞∞ |f(t)|² dt = (1 / 2π) ∫-∞∞ |F(ω)|² dω

Explanation:

• The total energy of the signal is preserved in both time and frequency domains.
• Important for signal energy and power calculations.

Example:

For f(t) = e-t², calculate energy in time domain:

E = ∫-∞∞ e-2t² dt = √(π/2)

In frequency domain:

|F(ω)|² = π e-ω²/2

So,

(1/2π) ∫-∞∞ π e-ω²/2 dω = (1/2) ∫-∞∞ e-ω²/2 dω = √(π/2) = E

Confirms energy preservation.

Summary Table of Properties

15 Solved Examples on Fourier Transform Properties

These examples range from basic to complex, aimed to develop complete mastery of Fourier Transform properties. Each example is clearly solved with step-by-step explanation. Mathematical symbols are explained along with the reasons for formulas used.

Example 1: Linearity of Fourier Transform

Problem: Find the Fourier Transform of: x(t) = 3cos(2t) + 2sin(5t)

Solution:

• Fourier Transform of cos(ω₀t) is π[δ(ω - ω₀) + δ(ω + ω₀)]
• Fourier Transform of sin(ω₀t) is jπ[δ(ω + ω₀) - δ(ω - ω₀)]

So:

FT{x(t)} =
3 × π[δ(ω - 2) + δ(ω + 2)] + 2 × jπ[δ(ω + 5) - δ(ω - 5)]

Example 2: Time Shifting

Problem: If f(t) = e-t² and its FT is F(ω) = √π e-ω²/4, find the FT of f(t - 2).

Solution: Time shift property: FT{f(t - t₀)} = e-jωt₀F(ω)

So:

FT{f(t - 2)} = e-j2ω √π e-ω²/4

Example 3: Frequency Shifting

Problem: Given f(t) has FT = F(ω), find FT of f(t)ej3t.

Solution: Frequency shift: FT{f(t)ejω₀t} = F(ω - ω₀)

FT{f(t)ej3t} = F(ω - 3)

Example 4: Scaling

Problem: If f(t) = e-t², find FT of f(2t).

Solution: Scaling property: FT{f(at)} = (1/|a|)F(ω/a)

FT{f(2t)} = (1/2) √π e-ω²/16

Example 5: Differentiation

Problem: If f(t) = e-t², find FT of f'(t)

Solution: FT{df/dt} = jωF(ω)

FT{f'(t)} = jω √π e-ω²/4

Example 6: Integration

Problem: Find FT of ∫-∞t e-τ² dτ

Solution: FT{∫ f(τ)dτ} = (1/jω)F(ω) + πF(0)δ(ω)

Example 7: Parseval’s Theorem

Problem: Verify Parseval’s theorem for f(t) = e-t²

Solution:

Time domain energy = ∫ e-2t² dt = √(π/2)

Frequency domain energy = (1/2π)∫ |√π e-ω²/4|² dω = √(π/2)

Example 8: Convolution Theorem

Problem: Given f(t) = e-t², g(t) = rect(t), find FT of f * g.

Solution:

FT{f * g} = F(ω) · G(ω)

F(ω) = √π e-ω²/4, G(ω) = sinc(ω/2)

Result: √π e-ω²/4 sinc(ω/2)

Example 9: Combination of Properties

Problem: Find FT of x(t) = 2e-(t-1)²cos(3t)

Solution:

• Use time shift, multiplication, and linearity.
• Step-by-step expansion required.

Not shown in full for brevity (can be done symbolically).

Example 10: Duality Principle

Problem: Show that if FT of f(t) = rect(t) is sinc(ω/2), then FT of sinc(t) is rect(ω/2π)

Example 11: Gaussian Transform

Problem: Prove FT of e-t² is √π e-ω²/4

Solution: Use integration techniques and FT definition.

Example 12: Time Reversal

Problem: FT of f(-t)

Solution: F{f(-t)} = F(-ω) (Mirror image in frequency)

Example 13: Even and Odd Decomposition

Problem: Decompose signal into even and odd parts and find FT

Solution: fe(t) = (f(t)+f(-t))/2, fo(t) = (f(t)-f(-t))/2

Example 14: System Response in Frequency Domain

Problem: Output of LTI system with impulse response h(t) and input x(t).

Solution: Y(ω) = H(ω)X(ω), then apply inverse FT

Example 15: Real World Application

Problem: In image processing, apply FT on image strip and analyze filtering effect.

Solution: Use 2D FT and apply high-pass filter in frequency domain, then inverse FT.

7 Solved Examples: Fourier Transform Properties

This section contains 20 well-explained, solved examples arranged from simple to advanced. Each example includes clear steps, explanations of formulas, and meaning of symbols used. These examples are meant to help learners fully understand and master the properties of the Fourier Transform.

Example 1: Linearity - Sum of Two Sine Waves

Given: f(t) = 3 sin(2t) + 2 sin(4t)

Task: Find Fourier Transform F(ω)

Solution:

We use the linearity property: F{a f(t) + b g(t)} = a F{f(t)} + b F{g(t)}

Fourier Transform of sin(at) = jπ[δ(ω + a) - δ(ω - a)]

So:

• F{sin(2t)} = jπ[δ(ω + 2) - δ(ω - 2)]
• F{sin(4t)} = jπ[δ(ω + 4) - δ(ω - 4)]

Then:

F(ω) = 3jπ[δ(ω + 2) - δ(ω - 2)] + 2jπ[δ(ω + 4) - δ(ω - 4)]

Example 2: Time Shifting

Given: f(t) = e^-tu(t), g(t) = f(t - 2)

Find: Fourier Transform of g(t)

Solution:

We use: F{f(t - t0)} = e^-jωt0F(ω)

Let F{f(t)} = 1 / (1 + jω)

Then:

F{g(t)} = e^-j2ω / (1 + jω)

Example 3: Frequency Shifting

Given: f(t) = cos(3t), g(t) = f(t)e^j2t

Find: Fourier Transform of g(t)

Solution:

Use: F{f(t)e^jω0t} = F(ω - ω0)

Let F{cos(3t)} = π[δ(ω + 3) + δ(ω - 3)]

So:

F{g(t)} = π[δ(ω - 1) + δ(ω - 5)]

Example 4: Scaling

Given: f(t) = e^-t2, find F{f(2t)}

Solution:

Use: F{f(at)} = (1/|a|) F(ω/a)

Let F{f(t)} = √π e^-ω2/4

Then:

F{f(2t)} = (1/2)√π e^-ω2/16

Example 5: Convolution Theorem

Given: f(t) = rect(t), g(t) = rect(t)

Find: Fourier Transform of f(t) * g(t)

Solution:

Use: F{f * g} = F(ω)·G(ω)

F{rect(t)} = sinc(ω/2)

So:

F{f * g} = sinc²(ω/2)

Example 6: Time Differentiation

Given: f(t) = e^-t2, find F{f'(t)}

Use: F{f'(t)} = jωF(ω)

Let F{f(t)} = √π e^-ω²/4

Then:

F{f'(t)} = jω√π e^-ω²/4

Example 7: Parseval’s Theorem

Given: f(t) = e^-t2, calculate energy

Solution:

Use: ∫|f(t)|²dt = (1/2π)∫|F(ω)|²dω

F(ω) = √π e^-ω²/4, so |F(ω)|² = π e^-ω²/2

Energy = (1/2π) ∫π e^-ω²/2dω = √(π/2)

Fourier Transform: Solved Examples

These examples help you fully understand the key properties of the Fourier Transform: linearity, time shifting, frequency shifting, scaling, convolution theorem, differentiation/integration, and Parseval's theorem. Each question includes full steps and explanations.

Example 1: Linearity

Problem: Find the Fourier Transform of f(t) = 3e^{-2t}u(t) + 2e^{-5t}u(t).

Solution: By linearity,

Let:

• F1(t) = e^{-2t}u(t) has FT F1(ω) = 1 / (2 + jω)
• F2(t) = e^{-5t}u(t) has FT F2(ω) = 1 / (5 + jω)

So:

Example 2: Time Shifting

Problem: Find the Fourier Transform of f(t) = e^{-t}u(t - 2).

Solution: Let g(t) = e^{-t}u(t) has FT G(ω) = 1 / (1 + jω)

Time shift by 2 → f(t) = g(t - 2) so FT becomes F(ω) = e^{-j2ω}G(ω)

F(ω) = e^{-j2ω} / (1 + jω)

Example 3: Frequency Shifting

Problem: FT of e^{j5t}e^{-2t}u(t)

Solution:

Let g(t) = e^{-2t}u(t) → FT is 1 / (2 + jω)

Multiplying by e^{j5t} shifts frequency:

Example 4: Scaling (Time)

Problem: FT of f(t) = e^{-2t}u(t) and f(2t)

Solution: Original FT is F(ω) = 1 / (2 + jω)

Then:

Example 5: Convolution Theorem

Problem: FT of convolution: f(t) = e^{-t}u(t) * e^{-2t}u(t)

Solution:

FT of first = 1 / (1 + jω), second = 1 / (2 + jω)

Key Explanation of Formula Symbols:

• ω (omega): Angular frequency (in radians/sec)
• j: Imaginary unit where j² = -1
• u(t): Unit step function (0 if t < 0, 1 if t ≥ 0)
• e^{-at}: Exponential decay (controls how fast signal dies out)
• *: Convolution operator

Why Use These Formulas?

Each formula represents how signals behave under certain conditions. With deep understanding, you can:

• Predict how filters affect signals
• Optimize signal processing systems
• Even design your own transforms tailored to special signal types

Tips to Master Fourier Transform:

• Practice breaking down signals into known components
• Understand how frequency and time are related (duality)
• Use theorems as tools to simplify problems

20 Solved Examples on Fourier Transform Properties

1. Linearity Property Example

Problem: Given f(t) = 2sin(t) and g(t) = 3cos(t), find Fourier Transform of h(t) = f(t) + g(t)

Solution: Using linearity:
F{f(t) + g(t)} = F{f(t)} + F{g(t)}
F{2sin(t)} = 2 × πj [δ(ω + 1) - δ(ω - 1)]
F{3cos(t)} = 3 × π [δ(ω + 1) + δ(ω - 1)]
Answer: H(ω) = 2πj[δ(ω + 1) - δ(ω - 1)] + 3π[δ(ω + 1) + δ(ω - 1)]

2. Time Shifting Example

Problem: Find Fourier Transform of f(t - 3), if F{f(t)} = F(ω)

Solution: Time shifting property:
F{f(t - t₀)} = e^-jωt₀F(ω)
So, F{f(t - 3)} = e^-j3ωF(ω)

3. Frequency Shifting Example

Problem: Find Fourier Transform of e^j5tf(t), if F{f(t)} = F(ω)

Solution: Frequency shifting:
F{e^jω₀tf(t)} = F(ω - ω₀)
So, F{e^j5tf(t)} = F(ω - 5)

4. Scaling Example

Problem: Find Fourier Transform of f(2t) if F{f(t)} = F(ω)

Solution: Scaling:
F{f(at)} = (1/|a|)F(ω/a)
So, F{f(2t)} = (1/2)F(ω/2)

5. Convolution Theorem Example

Problem: If F{f(t)} = F(ω), F{g(t)} = G(ω), find F{f * g}

Solution: Convolution theorem:
F{f(t) * g(t)} = F(ω)·G(ω)

6. Differentiation Example

Problem: Find Fourier Transform of d/dt[f(t)] if F{f(t)} = F(ω)

Solution:
F{df/dt} = jωF(ω)

7. Integration Example

Problem: Find Fourier Transform of ∫f(t)dt if F{f(t)} = F(ω)

Solution:
F{∫f(t)dt} = F(ω)/(jω) + πF(0)δ(ω)

8. Parseval's Theorem Example

Problem: Verify Parseval's theorem for f(t) = e^-t²

Solution:
Time domain: ∫|f(t)|² dt
Freq domain: (1/2π)∫|F(ω)|² dω
For f(t) = e^-t², both sides equal √π
So Parseval’s theorem holds

9. Use of Dirac Delta Example

Problem: Find Fourier Transform of δ(t)

Solution:
F{δ(t)} = 1 for all ω

10. Constant Function Example

Problem: f(t) = 1 for all t

Solution:
F{1} = 2πδ(ω)

11. Fourier Transform of Rectangular Pulse

Problem: f(t) = 1 for |t| ≤ 1/2

Solution:
F(ω) = ∫ from -1/2 to 1/2 e^-jωt dt = sinc(ω/2)

12. Fourier Transform of Exponential Function

Problem: f(t) = e^-atu(t), a > 0

Solution:
F(ω) = 1/(a + jω)

13. Sine Function

Problem: f(t) = sin(ω₀t)

Solution:
F(ω) = πj[δ(ω + ω₀) - δ(ω - ω₀)]

14. Cosine Function

Problem: f(t) = cos(ω₀t)

Solution:
F(ω) = π[δ(ω + ω₀) + δ(ω - ω₀)]

15. Step Function

Problem: f(t) = u(t)

Solution:
F(ω) = πδ(ω) + 1/(jω)

16. Impulse Response Analysis

Problem: Find F{δ(t - 2)}

Solution:
F(ω) = e^-j2ω

17. System Response using Convolution

Problem: If input = e^-tu(t), impulse response = δ(t), find output

Solution:
Output = input * h(t) = e^-tu(t)

18. Time Derivative and Spectral Magnitude

Problem: f(t) = t·e^-tu(t)

Solution:
F(ω) = 1/(a + jω)²

19. Modulated Signals

Problem: f(t) = cos(2πft)·e^-t²

Solution:
Use frequency shift and convolution theorem

20. Competent Case Study

Problem: Find Fourier Transform of signal f(t) = t²e^-t²

Solution:
Use integration by parts or table lookup.
F(ω) = √π(1 - ω²)e^-ω²/4
Each term shows time-domain energy affects frequency decay

Symbols and Meaning

• F(ω): Fourier Transform of f(t)
• ω: Angular frequency (rad/s)
• j: Imaginary unit (√-1)
• δ(t): Dirac delta function
• π: Constant (~3.14159)
• u(t): Unit step function