Convolution Theorem: Time & Frequency Domains

Hello! Welcome to your sixth lesson.

In our last lesson, we established a core toolkit of Fourier transform properties: linearity, shifting, scaling, and duality. These properties allow us to manipulate and understand signal spectra without repeatedly solving the transform integral. Today, we will build on that foundation to derive and interpret what is arguably the most operationally significant property in all of signal processing: the Convolution Theorem.

This theorem forms the fundamental link between a linear time-invariant (LTI) system's behavior in the time domain and its characteristics in the frequency domain. For an Electrical Engineering student like yourself, mastering this concept is the key to understanding filtering, modulation, and system analysis from a frequency-centric perspective.

Our learning outcome is to derive and interpret the convolution theorem in both time and frequency domains. We will see how this theorem simplifies complex problems and provides profound insight into how signals and systems interact.

1. What is Convolution?

Before we jump into the theorem, let's establish a firm understanding of the convolution operation itself. At its heart, convolution is a mathematical way to express the output of an LTI system given an input signal and the system's impulse response.

Conceptually, you can think of it as a sophisticated "weighted moving average." The output at any given time is a blend of the input signal's entire history, weighted by a characteristic function of the system.

To build this intuition, let's start with a resource that frames convolution from the practical problem of signal smoothing.

Convolution

This chapter from a textbook by Christopher Croke introduces convolution from the idea of using a moving average to reduce noise in a signal. This provides a practical motivation for the operation.

Please read the introductory part of the chapter, up to the formal definition (just before section 5.1). Focus on how the idea of a 'moving average' (Eq. 5.1) is generalized into a 'weighted moving average' (Eq. 5.3) using a weight function.

As the text shows, the weighted moving average leads to the formal definition of the convolution integral.

Definition: The convolution of two functions, $f (t)$ and $g (t)$ , is denoted by $(f * g) (t)$ and is defined as:

(f * g) (t) = \int_{- \infty}^{\infty} f (τ) g (t - τ) d τ

Here, $τ$ is a dummy variable of integration. The operation involves three steps for each output time $t$ :

Reflect: Take the function $g (τ)$ and reflect it about the vertical axis to get $g (- τ)$ .
Shift: Shift the reflected function by $t$ to get $g (t - τ)$ .
Integrate: Multiply this shifted-and-reflected function with $f (τ)$ and calculate the area under the product for all $τ$ .

This "flip-and-slide" process can be hard to visualize. The following resource provides excellent graphical and analytical examples to make it concrete.

The Convolution Operation

This Math LibreTexts page provides a step-by-step graphical and analytical walkthrough of convolving a box function with a triangle function.

Study 'Example 9.6.1' and 'Example 9.6.2'. Pay close attention to how the overlapping area changes as one function 'slides' past the other, and how this corresponds to the calculated value of the integral in different regions.

Convolution is also commutative, meaning $f * g = g * f$ . The proof is a simple change of variables ( $u = t - τ$ ):

(f * g) (t) = \int_{- \infty}^{\infty} f (τ) g (t - τ) d τ = \int_{\infty}^{- \infty} f (t - u) g (u) (- d u) = \int_{- \infty}^{\infty} g (u) f (t - u) d u = (g * f) (t)

2. The Time Convolution Theorem

Now we arrive at the main event. While convolution is a fundamental operation, calculating the integral directly can be difficult. The Convolution Theorem provides a powerful alternative by moving the problem into the frequency domain.

Theorem: Convolution in the time domain corresponds to multiplication in the frequency domain.
If $f (t) \leftrightarrow F (j ω)$ and $g (t) \leftrightarrow G (j ω)$ , then:

f (t) * g (t) \leftrightarrow F (j ω) G (j ω)

Proof:
To prove this, we take the Fourier transform of the convolution integral.

F {f (t) * g (t)} = \int_{t = - \infty}^{\infty} [\int_{τ = - \infty}^{\infty} f (τ) g (t - τ) d τ] e^{- j ω t} d t

Since the inner integral is with respect to $τ$ and the outer is with respect to $t$ , we can swap the order of integration (assuming the functions are well-behaved, which we do).

= \int_{τ = - \infty}^{\infty} f (τ) [\int_{t = - \infty}^{\infty} g (t - τ) e^{- j ω t} d t] d τ

Now, let's focus on the inner integral. It looks very similar to a Fourier transform. Using the time-shifting property we learned in the last lesson, $F {g (t - t_{0})} = e^{- j ω t_{0}} G (j ω)$ . Here, our time shift is $τ$ .
So, the inner integral is:

\int_{t = - \infty}^{\infty} g (t - τ) e^{- j ω t} d t = e^{- j ω τ} G (j ω)

Substituting this back into our main expression:

= \int_{τ = - \infty}^{\infty} f (τ) [e^{- j ω τ} G (j ω)] d τ

The term $G (j ω)$ does not depend on $τ$ , so we can pull it out of the integral.

= G (j ω) \int_{τ = - \infty}^{\infty} f (τ) e^{- j ω τ} d τ

The remaining integral is simply the definition of the Fourier transform of $f (t)$ , which is $F (j ω)$ .

= G (j ω) F (j ω) = F (j ω) G (j ω)

This completes the proof.

For a slightly different but equally valid approach to the proof, watch the following video.

The Fourier Transform and Convolution Integrals

This video by Prof. Steve Brunton proves the theorem in the reverse direction: by taking the inverse Fourier transform of the product of the spectra and showing that it results in the convolution integral. This reinforces the relationship from a different angle.

Watch the segment from 1:52 to 10:03. Follow the steps of substituting the transform definitions and rearranging the integrals. Don't worry if it seems complex; the key is to see how the convolution integral naturally emerges from the inverse transform of the product.

Interpretation: The Power of Spectral Multiplication

This theorem is far more than a mathematical convenience. It provides a new and profound way to interpret system behavior.

If $f (t)$ is the input signal to an LTI system and $h (t)$ is the system's impulse response, the output signal $y (t)$ is their convolution: $y (t) = f (t) * h (t)$ .

Applying the theorem, the output spectrum is:
$Y (j ω) = F (j ω) H (j ω)$

Here, $H (j ω)$ is the Fourier transform of the impulse response, known as the system's frequency response or transfer function. This equation tells us that an LTI system acts as a spectral filter. It modifies the input spectrum by multiplying it, frequency by frequency, with its own frequency response.

If $∣ H (j ω_{0}) ∣$ is large, the system amplifies the input signal's component at frequency $ω_{0}$ .
If $∣ H (j ω_{0}) ∣$ is small, the system attenuates that frequency component.
If $∣ H (j ω_{0}) ∣ = 0$ , the system blocks that frequency entirely.

The following video gives an excellent visual explanation of this filtering concept.

Convolution as spectral multiplication

In this video, Mike X Cohen demonstrates how convolution acts as a filter by examining the process in the frequency domain. This is key to interpreting the theorem.

Watch the clip from 4:23 to 9:59. Observe how the overlap (or lack thereof) between the signal's spectrum and the kernel's (filter's) spectrum determines the output. This is the core intuition behind filtering.

Test your understanding!

An ideal low-pass filter has a frequency response $H (j ω) = 1$ for $∣ ω ∣ \leq ω_{c}$ and $0$ otherwise. A signal $x (t) = cos (ω_{1} t) + cos (ω_{2} t)$ is passed through this filter, where $ω_{1} < ω_{c}$ and $ω_{2} > ω_{c}$ .

Without calculating any convolution integrals, what is the output signal $y (t)$ ?

Show answer

Transform the input: From the previous lesson, we know the Fourier transform of a cosine:
$X (j ω) = F {cos (ω_{1} t) + cos (ω_{2} t)} = π [δ (ω - ω_{1}) + δ (ω + ω_{1})] + π [δ (ω - ω_{2}) + δ (ω + ω_{2})]$
The input spectrum consists of four impulses at $\pm ω_{1}$ and $\pm ω_{2}$ .
Multiply in the frequency domain: The output spectrum is $Y (j ω) = X (j ω) H (j ω)$ . We multiply the input spectrum by the rectangular filter response.
- The impulses at $\pm ω_{1}$ are within the filter's passband ( $∣ \pm ω_{1} ∣ < ω_{c}$ ), so they are multiplied by 1 and passed through unchanged.
- The impulses at $\pm ω_{2}$ are outside the filter's passband ( $∣ \pm ω_{2} ∣ > ω_{c}$ ), so they are multiplied by 0 and eliminated.
Determine the output spectrum:
$Y (j ω) = π [δ (ω - ω_{1}) + δ (ω + ω_{1})]$
Inverse transform the output: This spectrum corresponds to the time-domain signal:
$y (t) = cos (ω_{1} t)$
The convolution theorem allowed us to determine that the filter simply removes the high-frequency component, a conclusion that would be much harder to reach by directly convolving $cos (ω_{1} t) + cos (ω_{2} t)$ with the $sinc$ function that is the impulse response of an ideal low-pass filter.

3. The Frequency Convolution Theorem (Duality)

Given the symmetry of the Fourier transform, it's natural to ask: what happens if we multiply in the time domain? As you might guess from the duality property, this corresponds to convolution in the frequency domain.

Theorem: Multiplication in the time domain corresponds to convolution in the frequency domain (with a scaling factor).
If $f (t) \leftrightarrow F (j ω)$ and $g (t) \leftrightarrow G (j ω)$ , then:

f (t) g (t) \leftrightarrow \frac{1}{2 π} [F (j ω) * G (j ω)]

Proof:
We can prove this directly by taking the Fourier transform of the product $f (t) g (t)$ and substituting the inverse transform definition for one of the functions (say, $f (t)$ ).

F {f (t) g (t)} = \int_{t = - \infty}^{\infty} f (t) g (t) e^{- j ω t} d t

Substitute $f (t) = \frac{1}{2 π} \int_{λ = - \infty}^{\infty} F (j λ) e^{j λ t} d λ$ :

= \int_{t = - \infty}^{\infty} [\frac{1}{2 π} \int_{λ = - \infty}^{\infty} F (j λ) e^{j λ t} d λ] g (t) e^{- j ω t} d t

Pull the $1/ (2 π)$ out and swap the order of integration:

= \frac{1}{2 π} \int_{λ = - \infty}^{\infty} F (j λ) [\int_{t = - \infty}^{\infty} g (t) e^{- j (ω - λ) t} d t] d λ

The inner integral is the Fourier transform of $g (t)$ evaluated at frequency $(ω - λ)$ , which is $G (j (ω - λ))$ .

= \frac{1}{2 π} \int_{λ = - \infty}^{\infty} F (j λ) G (j (ω - λ)) d λ

This final expression is, by definition, the convolution of $F (j ω)$ and $G (j ω)$ , scaled by $1/ (2 π)$ .

Interpretation: Modulation and Windowing

This dual theorem is the mathematical foundation of amplitude modulation (AM). If you have a baseband signal $f (t)$ (like audio) and multiply it by a carrier signal $g (t) = cos (ω_{c} t)$ , you are multiplying in the time domain.

The spectrum of the carrier is $G (j ω) = π [δ (ω - ω_{c}) + δ (ω + ω_{c})]$ .
The spectrum of the modulated signal, $f (t) cos (ω_{c} t)$ , is therefore:
$\frac{1}{2 π} [F (j ω) * G (j ω)] = \frac{1}{2 π} [F (j ω) * (π [δ (ω - ω_{c}) + δ (ω + ω_{c})])]$
$= \frac{1}{2} [F (j (ω - ω_{c})) + F (j (ω + ω_{c}))]$
Convolution with a delta function simply shifts the original function. So, multiplying by a cosine in time convolves the signal's spectrum with two delta functions, creating two copies of the original spectrum centered at the carrier frequencies $\pm ω_{c}$ . This is precisely how radio communication shifts a low-frequency signal to a high-frequency band for transmission.

Conclusion

Today, we've explored one of the most pivotal concepts in signal processing. The convolution theorem is not just a calculation trick; it's a new lens through which to view the interaction of signals and systems.

Key Takeaways:

Convolution in Time = Multiplication in Frequency: This is the principle behind filtering. An LTI system's effect is to multiply the input signal's spectrum by the system's frequency response.
Multiplication in Time = Convolution in Frequency: This is the principle behind modulation. Multiplying a signal by a sinusoid shifts its entire spectrum to a new center frequency.
These two theorems form a dual pair that is fundamental to analyzing, designing, and understanding signal processing systems.

Preview of the Next Lesson:
We have one more major theorem to cover in this module: Parseval's Theorem. This theorem will connect the energy of a signal in the time domain to its energy in the frequency domain, providing a crucial tool for power and energy analysis in signals and systems.

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