Fourier Transform Properties: Proofs and Applications

Hello! Welcome to your fifth lesson.

In our last session, we took a major step by deriving the Fourier transform from the Fourier series. We saw how, by letting the period of a signal approach infinity, the discrete spectral lines of the Fourier series merge into a continuous spectrum. This gave us the Fourier transform pair, our tool for analyzing the frequency content of non-periodic signals.

However, calculating the Fourier transform for every new signal by solving the integral can be tedious and doesn't always yield the best insights. The real power of the transform comes from understanding its fundamental properties. These properties are the "rules of the game" that allow us to predict, analyze, and manipulate signals and their spectra with far greater ease and intuition.

Today's learning outcome is to prove and apply key Fourier transform properties including linearity, time-shifting, frequency-shifting, scaling, and duality. Mastering these will form the bedrock of your ability to think and work in the frequency domain.

A Quick Recap of Key Definitions

As a reminder, here is the Fourier transform pair we are working with:

Forward Transform (Analysis): $X (j ω) = \int_{- \infty}^{\infty} x (t) e^{- j ω t} d t$
Inverse Transform (Synthesis): $x (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} X (j ω) e^{j ω t} d ω$

We'll use the notation $x (t) \leftrightarrow X (j ω)$ to denote a Fourier transform pair.

1. Linearity

The linearity property is the most fundamental. It states that the Fourier transform of a linear combination of signals is the same linear combination of their individual Fourier transforms.

If $x_{1} (t) \leftrightarrow X_{1} (j ω)$ and $x_{2} (t) \leftrightarrow X_{2} (j ω)$ , then for any constants $a$ and $b$ :

a x_{1} (t) + b x_{2} (t) \leftrightarrow a X_{1} (j ω) + b X_{2} (j ω)

Proof:
The proof follows directly from the linearity of integration.

F {a x_{1} (t) + b x_{2} (t)} = \int_{- \infty}^{\infty} [a x_{1} (t) + b x_{2} (t)] e^{- j ω t} d t

= a \int_{- \infty}^{\infty} x_{1} (t) e^{- j ω t} d t + b \int_{- \infty}^{\infty} x_{2} (t) e^{- j ω t} d t

= a X_{1} (j ω) + b X_{2} (j ω)

This property is crucial because it allows us to decompose complex signals into simpler parts, transform them individually, and then reassemble the results.

The following resource provides a quick overview of several properties we'll discuss. For now, let's focus on the linearity property.

Lecture 9, Fourier Transform Properties | MIT RES.6.007 Signals and Systems, Spring 2011

This lecture by Prof. Alan Oppenheim from MIT OpenCourseWare is a classic resource. We'll be watching several clips from it. This first short segment confirms the linearity property.

Watch the clip from 22:03 to 22:32.

2. Time and Frequency Shifting

Shifting a signal is a common operation. How does it affect the spectrum?

Time-Shifting

A shift in the time domain by $t_{0}$ does not alter the magnitude of the frequency spectrum, but it introduces a linear phase shift in the frequency domain.

Property:

x (t - t_{0}) \leftrightarrow e^{- j ω t_{0}} X (j ω)

Proof:
Let's compute the transform of the shifted signal $x (t - t_{0})$ . We use a change of variables, letting $τ = t - t_{0}$ , which means $t = τ + t_{0}$ and $d t = d τ$ .

F {x (t - t_{0})} = \int_{- \infty}^{\infty} x (t - t_{0}) e^{- j ω t} d t

= \int_{- \infty}^{\infty} x (τ) e^{- j ω (τ + t_{0})} d τ

= \int_{- \infty}^{\infty} x (τ) e^{- j ω τ} e^{- j ω t_{0}} d τ

Since $e^{- j ω t_{0}}$ is constant with respect to $τ$ , we can pull it out of the integral:

= e^{- j ω t_{0}} \int_{- \infty}^{\infty} x (τ) e^{- j ω τ} d τ = e^{- j ω t_{0}} X (j ω)

The magnitude of the term $e^{- j ω t_{0}}$ is always 1, so only the phase is affected. This makes intuitive sense: delaying a signal shouldn't change the frequencies present in it, but it should change their relative alignment in time (their phase).

Frequency-Shifting (Modulation)

Symmetrically, multiplying a signal by a complex exponential in the time domain causes a shift in the frequency domain. This is the mathematical basis for amplitude modulation (AM).

Property:

e^{j ω_{0} t} x (t) \leftrightarrow X (j (ω - ω_{0}))

Proof:
The proof is structurally identical to the time-shifting proof.

F {e^{j ω_{0} t} x (t)} = \int_{- \infty}^{\infty} [e^{j ω_{0} t} x (t)] e^{- j ω t} d t

= \int_{- \infty}^{\infty} x (t) e^{- j (ω - ω_{0}) t} d t

This is precisely the definition of the Fourier transform of $x (t)$ , but evaluated at the frequency $(ω - ω_{0})$ .

= X (j (ω - ω_{0}))

This property is powerful. It tells us we can shift the entire spectrum of a signal to a different frequency band by simply multiplying it by a sinusoid. This is how a low-frequency audio signal can be moved to a high-frequency radio-carrier band for transmission.

The following video segment explains the time-shifting property.

Lecture 9, Fourier Transform Properties | MIT RES.6.007 Signals and Systems, Spring 2011

Let's return to the MIT lecture to get an explanation of the time-shifting property.

Watch the clip from 18:58 to 20:01. The key takeaway is the correspondence between a time delay and a linear phase shift.

Test your understanding!

The Fourier transform of a rectangular pulse $x (t) = 1$ for $∣ t ∣ \leq T_{1}$ and 0 otherwise is $X (j ω) = \frac{2 s i n ( ω T _{1} )}{ω}$ .

Using the properties you just learned, what is the Fourier transform of $y (t) = cos (ω_{0} t)$ ?

Hint: Use Euler's formula, $cos (θ) = \frac{1}{2} (e^{j θ} + e^{- j θ})$ , and the Fourier transform of $x (t) = 1$ , which is $X (j ω) = 2 π δ (ω)$ .

Show answer

Let's represent $cos (ω_{0} t)$ using Euler's formula:
$y (t) = cos (ω_{0} t) = \frac{1}{2} e^{j ω_{0} t} + \frac{1}{2} e^{- j ω_{0} t}$
Let's consider the signal $x (t) = 1$ . Its Fourier transform is $X (j ω) = 2 π δ (ω)$ .

Now we can apply linearity and the frequency-shifting property.
The term $\frac{1}{2} e^{j ω_{0} t}$ is $\frac{1}{2} e^{j ω_{0} t} \cdot x (t)$ . Its transform is:
$F {\frac{1}{2} e^{j ω_{0} t} \cdot 1} = \frac{1}{2} X (j (ω - ω_{0})) = \frac{1}{2} [2 π δ (ω - ω_{0})] = π δ (ω - ω_{0})$

The term $\frac{1}{2} e^{- j ω_{0} t}$ is $\frac{1}{2} e^{j (- ω_{0}) t} \cdot x (t)$ . Its transform is:
$F {\frac{1}{2} e^{- j ω_{0} t} \cdot 1} = \frac{1}{2} X (j (ω - (- ω_{0}))) = \frac{1}{2} [2 π δ (ω + ω_{0})] = π δ (ω + ω_{0})$

By linearity, the transform of the sum is the sum of the transforms:
$Y (j ω) = π δ (ω - ω_{0}) + π δ (ω + ω_{0})$
This shows that a pure cosine wave has a spectrum consisting of two impulses at its positive and negative frequencies, which is a fundamental result in signal processing.

3. Time Scaling

This property reveals the inverse relationship between a signal's duration and its bandwidth. Compressing a signal in time causes its spectrum to expand, and vice-versa.

Property:
For a real constant $a \neq = 0$ :

x (a t) \leftrightarrow \frac{1}{∣ a ∣} X (j \frac{ω}{a})

Proof:
Again, we use a change of variables, $τ = a t$ . This means $t = τ / a$ and $d t = d τ / a$ .
Let's first consider $a > 0$ .

F {x (a t)} = \int_{- \infty}^{\infty} x (a t) e^{- j ω t} d t = \int_{- \infty}^{\infty} x (τ) e^{- j ω (τ / a)} \frac{d τ}{a}

= \frac{1}{a} \int_{- \infty}^{\infty} x (τ) e^{- j (ω / a) τ} d τ = \frac{1}{a} X (j \frac{ω}{a})

If $a < 0$ , the limits of integration are flipped, which introduces a negative sign. This is absorbed by the absolute value $∣ a ∣$ . This inverse relationship is a manifestation of a principle similar to the Heisenberg Uncertainty Principle: a signal cannot be arbitrarily localized in both time and frequency.

The following video provides an excellent intuitive demonstration of this property.

Lecture 9, Fourier Transform Properties | MIT RES.6.007 Signals and Systems, Spring 2011

Prof. Oppenheim provides a very memorable demonstration of the time-scaling property using a glockenspiel. This illustrates the inverse relationship between time and frequency in a practical way.

Watch the segment from 5:34 to 11:42. Observe how playing a recorded note at half speed (stretching time) lowers its pitch by an octave (compressing frequency), and playing it at double speed does the opposite.

4. Duality

Duality is arguably the most elegant property of the Fourier transform. It arises from the strong similarity between the forward and inverse transform integrals. It essentially says that if a particular functional shape appears in the time domain, its corresponding transform shape can appear in the frequency domain, and vice-versa.

Property:
If $x (t) \leftrightarrow X (j ω)$ , then:

X (j t) \leftrightarrow 2 π x (- ω)

This means that if we take the function for the Fourier transform, $X (j ω)$ , and treat it as a time-domain signal by replacing $ω$ with $t$ , its Fourier transform will be the original time-domain function, but time-reversed and scaled by $2 π$ .

The following video provides a very clear, step-by-step mathematical proof of this property.

Duality Property of Fourier Transform

This video from Neso Academy focuses solely on proving the duality property. Your preference for mathematical depth will be well-served here.

Watch from the beginning to 4:26. Follow the manipulation of the inverse transform integral to derive the duality relationship.

Let's re-state the proof from the video using our course's notation to ensure consistency.

Proof of Duality:

Start with the inverse transform definition:
$x (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} X (j ω) e^{j ω t} d ω$
Multiply both sides by $2 π$ :
$2 π x (t) = \int_{- \infty}^{\infty} X (j ω) e^{j ω t} d ω$
Replace all instances of $t$ with $- t$ :
$2 π x (- t) = \int_{- \infty}^{\infty} X (j ω) e^{- j ω t} d ω$
Now, let's interchange the roles of the variables $t$ and $ω$ . Replace every $t$ with $ω$ and every $ω$ with $t$ :
$2 π x (- ω) = \int_{- \infty}^{\infty} X (j t) e^{- j ω t} d t$
Look closely at the right-hand side. It is precisely the definition of the forward Fourier transform of the signal $X (j t)$ . Therefore:
$F {X (j t)} = 2 π x (- ω)$

A classic example of duality is the relationship between a rectangular pulse in time and the sinc function in frequency.

Pair 1: A rectangular pulse in time, $rect (t)$ , has a $sinc (ω)$ shaped Fourier transform.
Duality implies Pair 2: A $sinc (t)$ function in time must have a rectangular pulse shaped Fourier transform.

This "two for one" deal is incredibly useful for finding new transform pairs without integration.

Summary of Properties

Here is a table summarizing the properties we've covered. This is a crucial reference for signal analysis.

Property Name	Time Domain $x (t)$	Frequency Domain $X (j ω)$
Linearity	$a x_{1} (t) + b x_{2} (t)$	$a X_{1} (j ω) + b X_{2} (j ω)$
Time-Shifting	$x (t - t_{0})$	$e^{- j ω t_{0}} X (j ω)$
Frequency-Shifting	$e^{j ω_{0} t} x (t)$	$X (j (ω - ω_{0}))$
Time Scaling	$x (a t)$	$\frac{1}{
Duality	$X (j t)$	$2 π x (- ω)$

This table is largely consistent with Table 4.2 in the Princeton course notes you can find here.

Conclusion

Today, we've moved beyond the basic definition of the Fourier transform and equipped ourselves with a powerful set of tools for analysis. We have formally proven and intuitively explored the properties of linearity, time- and frequency-shifting, scaling, and duality.

Key Takeaways:

Linearity allows us to use a "divide and conquer" strategy for complex signals.
Shifting in one domain corresponds to multiplication by a complex exponential (a phase rotation) in the other.
Scaling reveals the fundamental trade-off: compressing a signal in time expands its frequency spectrum, and vice-versa.
Duality is an elegant symmetry that allows us to find new transform pairs from existing ones, effectively doubling our knowledge.

Preview of the Next Lesson:
With these properties in hand, we are now ready to tackle one of the most important results in all of signal processing: the Convolution Theorem. We will derive this theorem, which states that convolution in the time domain becomes simple multiplication in the frequency domain. This is the key that unlocks the analysis of Linear Time-Invariant (LTI) systems using Fourier methods and is the foundation of filtering.

Recommended resources

A Quick Recap of Key Definitions

1. Linearity

2. Time and Frequency Shifting

Time-Shifting

Frequency-Shifting (Modulation)

3. Time Scaling

4. Duality

Summary of Properties

Conclusion