How Does Fourier Transform Reveal Frequency?

 

Signals and Systems | Fourier Transform

Does x(t)e^-j2πft Reveal Frequency? Understanding the Fourier Transform

The expression x(t)e^-j2πft is an important part of the Fourier Transform. It is used to analyse the frequency components present in a continuous-time signal. However, this expression alone does not reveal the complete frequency spectrum. We must integrate it over time to obtain the Fourier Transform X(f).

1. What Is the Fourier Transform?

The Fourier Transform is a mathematical operation that converts a signal from the time domain into the frequency domain. It helps us understand which frequencies are present in a signal and their corresponding magnitude and phase.

Continuous-Time Fourier Transform Formula

X(f) = ∫_−∞^∞ x(t)e^−j2πft dt

Meaning of Each Term

Symbol	Meaning
x(t)	Original time-domain signal
e−j2πft	Complex exponential frequency-analysis kernel
f	Frequency measured in Hertz (Hz)
t	Time measured in seconds
X(f)	Frequency-domain representation

2. What Does e^−j2πft Actually Do?

To understand the Fourier Transform, we use Euler's formula. It expresses a complex exponential in terms of sine and cosine.

e^−j2πft = cos(2πft) − j sin(2πft)

Cosine Component

Represents the real part of the complex exponential.

Sine Component

Represents the imaginary part of the complex exponential.

The Fourier Transform compares the original signal with these complex sinusoidal components at different frequencies. Integration determines the contribution of each frequency.

3. Fourier Transform Solved Example

Consider the following continuous-time signal:

x(t) = e^−2t,   t ≥ 0

Step 1: Apply the Fourier Transform Formula

X(f) = ∫₀^∞ e^−2te^−j2πft dt

Step 2: Combine the Exponents

X(f) = ∫₀^∞ e^{−(2+j2πf)t} dt

X(f) = [ −e^{−(2+j2πf)t} / (2+j2πf) ]₀^∞

Step 3: Apply the Integration Limits

At t → ∞, the exponential term approaches zero.

At t = 0, the exponential term equals one.

X(f) = 1 / (2+j2πf)

Final Answer

X(f) = 1 / (2+j2πf)

This is the frequency-domain representation of the given exponential signal.

4. Difference Between x(t), X(f), and |X(f)|²

Expression	Meaning
x(t)	Time-domain signal
X(f)	Complex Fourier Transform
|X(f)|	Magnitude spectrum
|X(f)|²	Energy spectral density for an energy signal

5. Fourier Transform vs Inverse Fourier Transform

Fourier Transform

Time Domain → Frequency Domain

Converts x(t) into X(f).

Inverse Fourier Transform

Frequency Domain → Time Domain

Converts X(f) back into x(t).

x(t) ⇌ X(f)

Fourier Transform Pair

Important Exam Tips: Fourier Transform

• Fourier Transform converts a time-domain signal into a frequency-domain signal.
• The complex exponential e^−j2πft is the frequency-analysis kernel.
• Frequency f is measured in Hertz (Hz).
• |X(f)| represents the magnitude spectrum.
• |X(f)|² represents energy spectral density for an energy signal.
• Inverse Fourier Transform converts the frequency-domain signal back into the time domain.

Assume bandwidth B Hz contains 99% of the total energy.

Let's understand the calculation carefully.

1. What does bandwidth B mean here?

The question says:

"99% of the energy of x(t) lies within B Hz."

Since the signal is real, its energy spectrum is symmetric around zero frequency.

Therefore, the frequency range is:

$$\boxed{-B\leq f\leq+B}$$

Energy distribution

Inside −B to +B: 99%

Outside: 1%

Illustrative energy spectrum, not to scale.

2. Calculate total energy

Given:

$$x(t)=e^{-2t}$$

Total energy:

$$E=\int_0^\infty |x(t)|^2dt$$

$$=\int_0^\infty e^{-4t}dt$$

$$\boxed{E=\frac14}$$

3. Calculate 99% energy

Since B contains 99% of the energy:

$$E_B=0.99\times E$$

$$E_B=0.99\times\frac14$$

$$\boxed{E_B=0.2475}$$

This is the energy that must be present between −B and +B.

4. Apply Fourier Transform

We already calculated:

$$X(f)=\frac{1}{2+j2\pi f}$$

Therefore:

$$|X(f)|^2= \frac{1}{4(1+\pi^2f^2)}$$

Using Parseval's theorem:

$$\int_{-B}^{B} \frac{1}{4(1+\pi^2f^2)}df =0.2475$$

Integrating:

$$\frac{1}{2\pi}\tan^{-1}(\pi B)=0.2475$$

$$\tan^{-1}(\pi B)=0.495\pi$$

Taking tangent:

$$\pi B=\tan(0.495\pi)$$

$$B=\frac{\tan(0.495\pi)}{\pi}$$

$$\boxed{{\displaystyle B\approx 20.26\text{ Hz}}}$$

Important: What if B contains 100% energy?

This is a very important conceptual difference.

Condition	Required bandwidth
99% energy	Approximately 20.26 Hz
100% energy	Infinite bandwidth

Why infinite?

Because the energy spectrum extends over all frequencies. It approaches zero as frequency increases, but never becomes exactly zero at any finite frequency.

Remember this for exams:

• 99% energy → finite bandwidth.

• 100% energy → infinite bandwidth for this signal.

That is why we use the 99% energy condition rather than trying to capture all the energy.

Frequently Asked Questions (FAQs)

Q1. Does e^−j2πft represent frequency?

It is a complex exponential associated with frequency f. It acts as the frequency-analysis kernel in the Fourier Transform.

Q2. What is the purpose of the Fourier Transform?

It identifies the frequency components of a signal and describes their magnitude and phase.

Q3. What is the difference between time domain and frequency domain?

The time domain describes how a signal changes with time, whereas the frequency domain describes its frequency components.

Q4. What is the Fourier Transform of e^−2tu(t)?

X(f) = 1/(2+j2πf), where u(t) is the unit step function.

Conclusion

The expression x(t)e^−j2πft is a fundamental component of the Fourier Transform. By integrating this expression over time, we obtain X(f), which describes the frequency-domain characteristics of the original signal. Understanding this concept is essential for Signals and Systems, Digital Signal Processing (DSP), and engineering competitive examinations.