Signals and Systems

4.5 The Multiplication Property

Imron Rosyadi

The Multiplication Property

4.5 The Multiplication Property

The multiplication property is the dual of the convolution property. It states that multiplication in the time domain corresponds to convolution in the frequency domain.

\[ \begin{equation*} r(t)=s(t) p(t) \longleftrightarrow R(j \omega)=\frac{1}{2 \pi} \int_{-\infty}^{+\infty} S(j \theta) P(j(\omega-\theta)) d \theta \tag{4.70} \end{equation*} \]

This property is crucial for understanding amplitude modulation, a core concept in communication systems.

Note

Duality Principle:
Many Fourier Transform properties have a dual. If an operation in one domain corresponds to another operation in the dual domain, then the inverse also holds (with scaling factors).

Just as convolution in time became multiplication in frequency, multiplication in time becomes convolution in frequency. This duality is a powerful concept in Fourier analysis.
Equation 4.70 shows this formally. The integral is a convolution integral, but it’s performed in the frequency domain. The \(1/(2\pi)\) factor is important and arises from the definition of the Fourier Transform and its inverse.
This property is sometimes called the “modulation property” because multiplication by a carrier signal (often a sinusoid) is the basis of amplitude modulation. This is how we shift signals to different frequency bands for transmission.

Example 4.21: Amplitude Modulation

Let \(s(t)\) be a signal with spectrum \(S(j \omega)\). Consider multiplying it by a sinusoidal carrier \(p(t)=\cos \omega_{0} t\).

The Fourier Transform of \(p(t)\) is \(P(j \omega)=\pi \delta\left(\omega-\omega_{0}\right)+\pi \delta\left(\omega+\omega_{0}\right)\).

Applying the multiplication property, the spectrum \(R(j \omega)\) of \(r(t)=s(t) p(t)\) is:

\[ \begin{align*} R(j \omega) & =\frac{1}{2 \pi} \int_{-\infty}^{+\infty} S(j \theta) P(j(\omega-\theta)) d \theta \\ & =\frac{1}{2} S\left(j\left(\omega-\omega_{0}\right)\right)+\frac{1}{2} S\left(j\left(\omega+\omega_{0}\right)\right) \tag{4.71} \end{align*} \]

The original signal’s information is preserved but shifted to higher frequencies.
This is the basis of sinusoidal amplitude modulation.

This example is fundamental to understanding how communication systems work.
We take a baseband signal \(s(t)\), which typically occupies low frequencies (e.g., speech or audio).
We multiply it by a high-frequency cosine wave, the carrier \(p(t)=\cos(\omega_0 t)\).
In the frequency domain, this multiplication translates to convolving \(S(j\omega)\) with the two impulses of \(P(j\omega)\).
The result is that the spectrum \(S(j\omega)\) is split into two copies, each scaled by 1/2, and shifted to be centered around \(+\omega_0\) and \(-\omega_0\).
This process allows us to transmit signals over radio waves or other high-frequency channels. We assume here that \(\omega_0\) is large enough so that the shifted copies of \(S(j\omega)\) do not overlap. If they overlap, distortion occurs.

Interactive Amplitude Modulation

Adjust the carrier frequency \(\omega_0\) and the bandwidth of the message signal \(\omega_1\).
Observe how the message spectrum \(S(j\omega)\) is shifted and replicated.

viewof omega0_mod = Inputs.range([5, 20], {step: 1, value: 10, label: "Carrier Frequency (ω0)"})
viewof omega1_mod = Inputs.range([1, 4], {step: 0.5, value: 2, label: "Message Bandwidth (ω1)"})

Example 4.22: Demodulation

To recover the original signal \(s(t)\) from the modulated signal \(r(t)\), we can use demodulation.
Multiply \(r(t)\) by the same carrier \(p(t)=\cos \omega_{0} t\) again: \(g(t)=r(t) p(t)\).

\(G(j \omega)\) will contain a scaled version of \(S(j \omega)\) at baseband, plus components shifted to \(\pm 2 \omega_{0}\).
Specifically, \(G(j\omega) = \frac{1}{2} S(j\omega) + \frac{1}{4}S(j(\omega-2\omega_0)) + \frac{1}{4}S(j(\omega+2\omega_0))\).

A frequency-selective lowpass filter can then extract the original signal.

Tip

This process of modulation and demodulation is fundamental to wireless communication, allowing multiple signals to share the same medium by occupying different frequency bands.

After modulating \(s(t)\) to \(r(t)\), the signal is at a higher frequency. To listen to it (e.g., on a radio), we need to bring it back to baseband. This is demodulation.
Multiplying \(r(t)\) by \(\cos(\omega_0 t)\) again effectively shifts the spectrum \(R(j\omega)\) both to the left and right by \(\omega_0\).
The component of \(R(j\omega)\) centered at \(\omega_0\) shifts to \(0\) and \(2\omega_0\).
The component of \(R(j\omega)\) centered at \(-\omega_0\) shifts to \(-2\omega_0\) and \(0\).
The terms shifting to \(0\) add up to form \(\frac{1}{2}S(j\omega)\) at baseband. The terms shifting to \(\pm 2\omega_0\) are unwanted high-frequency components.
A lowpass filter is then used to remove these high-frequency components, leaving only the desired \(s(t)\). This is how a radio receiver works.

Interactive Demodulation and Filtering

Continue from the previous modulation example. We now multiply \(r(t)\) by \(p(t)\) again to get \(g(t)\), and then apply a lowpass filter.

viewof omega0_demod = Inputs.range([5, 20], {step: 1, value: 10, label: "Carrier Frequency (ω0)"})
viewof omega1_demod = Inputs.range([1, 4], {step: 0.5, value: 2, label: "Message Bandwidth (ω1)"})
viewof lp_cutoff = Inputs.range([1, 10], {step: 0.5, value: 3, label: "Lowpass Filter Cutoff"})

Example 4.23: Fourier Transform of a Product

Determine the Fourier Transform of \(x(t)=\frac{\sin (t) \sin (t / 2)}{\pi t^{2}}\).

Recognize \(x(t)\) as a product of two sinc functions:
\(x(t)=\pi\left(\frac{\sin (t)}{\pi t}\right)\left(\frac{\sin (t / 2)}{\pi t}\right)\)

Recall \(\mathcal{F}\left\{\frac{\sin W t}{\pi t}\right\} = \text{rect}(\omega/2W)\).
So, \(\mathcal{F}\left\{\frac{\sin (t)}{\pi t}\right\}\) is a rect pulse of width 2 (from -1 to 1).
And \(\mathcal{F}\left\{\frac{\sin (t / 2)}{\pi t}\right\}\) is a rect pulse of width 1 (from -0.5 to 0.5).

Applying the multiplication property:
\(X(j \omega)=\frac{1}{2 \pi} \mathcal{F}\left\{\frac{\sin (t)}{\pi t}\right\} * \mathcal{F}\left\{\frac{\sin (t / 2)}{\pi t}\right\}\)

The convolution of two rectangular pulses results in a triangular pulse.

This example demonstrates a practical application of the multiplication property for finding the Fourier Transform of a somewhat complex signal.
Instead of directly integrating the Fourier Transform definition for \(x(t)\), we leverage known Fourier Transform pairs and properties.
The key insight is to recognize the sinc functions. The Fourier Transform of \(\frac{\sin(Wt)}{\pi t}\) is a rectangular pulse from \(-W\) to \(W\) with amplitude 1.
For \(\frac{\sin(t)}{\pi t}\), \(W=1\), so its FT is a rect from -1 to 1.
For \(\frac{\sin(t/2)}{\pi t}\), the argument is \(t/2\), so \(W=1/2\). Its FT is a rect from -0.5 to 0.5.
Convolving these two rectangular pulses in the frequency domain gives a triangular pulse shape. The \(1/(2\pi)\) factor comes from the multiplication property.

Interactive Convolution of Rectangular Pulses

Visualize the convolution of two rectangular pulses in the frequency domain.

4.5.1 Frequency-Selective Filtering with Variable Center Frequency

The multiplication property enables the creation of tunable frequency-selective filters.
Instead of physically changing filter components, we shift the signal’s spectrum.

Consider the system below for implementing a bandpass filter:

graph LR
    A["x(t)"] --> B{"x(t) * e^(jωct)"}
    B --> C["y(t)"]
    C --> D("Lowpass Filter, H_LP(jω)")
    D --> E["w(t)"]
    E --> F{"w(t) * e^(-jωct)"}
    F --> G["f(t)"]

Traditional analog filters have fixed characteristics determined by their components (resistors, capacitors, inductors). To change the center frequency of a bandpass filter, you’d need to change multiple components simultaneously, which is difficult. This section introduces a clever technique using modulation to achieve a tunable filter without physically altering the filter itself. The core idea is to shift the desired frequency band down to baseband, filter it with a fixed lowpass filter, and then shift it back to the desired center frequency. The Mermaid diagram illustrates the block diagram for this system.

Spectral Transformation in Tunable Filtering

Let’s trace the spectrum of the signal through the system:

1. Input \(x(t)\): Has spectrum \(X(j\omega)\).
2. Multiply by \(e^{j \omega_{c} t}\): \(y(t) = x(t)e^{j \omega_{c} t}\).
   Using the frequency-shifting property, \(Y(j \omega) = X(j(\omega-\omega_{c}))\).
   The spectrum of \(x(t)\) is shifted to the right by \(\omega_c\).
3. Lowpass Filter: \(w(t)\) has spectrum \(W(j\omega) = Y(j\omega)H_{LP}(j\omega)\).
   This selects the portion of \(Y(j\omega)\) that falls within the lowpass filter’s bandwidth.
4. Multiply by \(e^{-j \omega_{c} t}\): \(f(t) = w(t)e^{-j \omega_{c} t}\).
   The spectrum \(W(j\omega)\) is shifted to the left by \(\omega_c\).

Important

By varying \(\omega_c\), the center frequency of the effective bandpass filter changes, without altering the lowpass filter.

This slide walks through the spectral operations at each stage of the tunable filter system.
The first multiplication by \(e^{j\omega_c t}\) is key. It takes the baseband signal \(X(j\omega)\) and shifts its entire spectrum up by \(\omega_c\). This means the frequencies around \(0\) in \(X(j\omega)\) are now around \(\omega_c\) in \(Y(j\omega)\).
The fixed lowpass filter then acts on this shifted spectrum. It effectively “picks out” a band of frequencies from the shifted signal that corresponds to the original signal’s passband, but now centered around \(\omega_c\).
The final multiplication by \(e^{-j\omega_c t}\) shifts this selected band back down. The part that was around \(\omega_c\) and passed by the lowpass filter is now shifted back to baseband, and the part that was around \(0\) in \(W(j\omega)\) is now around \(-\omega_c\).
The net effect is that the system acts as a bandpass filter whose center frequency is determined by \(\omega_c\).

Interactive Tunable Bandpass Filter

Adjust the carrier frequency \(\omega_c\) and the lowpass filter cutoff \(\omega_0\).
Observe how the spectrum evolves through the system.

viewof omega_c_tunable = Inputs.range([5, 20], {step: 1, value: 10, label: "Carrier Frequency (ωc)"})
viewof lp_cutoff_tunable = Inputs.range([1, 4], {step: 0.5, value: 2, label: "Lowpass Filter Cutoff (ω0)"})
viewof message_bw_tunable = Inputs.range([0.5, 2], {step: 0.1, value: 1, label: "Message Bandwidth (Wm)"})

Equivalent Bandpass Filter and Real Signals

The overall system effectively creates a bandpass filter.

• For complex signals, the system of Figure 4.26 is equivalent to an ideal bandpass filter with center frequency \(-\omega_c\) and bandwidth \(2\omega_0\).
• By varying \(\omega_c\), we tune the center frequency of this bandpass filter.

If \(x(t)\) is a real signal, the intermediate signals \(y(t), w(t), f(t)\) are generally complex.
If we take only the real part of \(f(t)\), the resulting spectrum will be symmetric, passing bands of frequencies centered around \(\omega_c\) and \(-\omega_c\).

The first point on this slide summarizes the complex exponential modulation case. The system effectively creates a “negative” center frequency bandpass filter when using \(e^{j\omega_c t}\) and \(e^{-j\omega_c t}\).
The second column addresses the more common scenario where the input signal \(x(t)\) is real. If \(x(t)\) is real, then its spectrum \(X(j\omega)\) is conjugate symmetric. When you multiply by \(e^{j\omega_c t}\), you get a single-sided spectrum. However, if you use a real carrier like \(\cos(\omega_c t)\) for both modulation and demodulation (as in Example 4.22), then the spectrum of the output will be symmetric (it will have both positive and negative frequency components). This effectively means the filter passes both positive and negative frequency bands. This is a common setup in practical communication systems.

4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs

The Fourier Transform properties and common transform pairs are invaluable tools for signal and system analysis.

Key Properties (Table 4.1):

• Linearity
• Time Shift
• Frequency Shift
• Conjugation
• Time Reversal
• Differentiation/Integration
• Scaling
• Convolution in Time <-> Multiplication in Frequency
• Multiplication in Time <-> Convolution in Frequency
• Duality
• Parseval’s Relation

Basic Transform Pairs (Table 4.2):

• Impulses (\(\delta(t)\), \(\delta(\omega)\))
• Exponentials (\(e^{j\omega_0 t}\), \(e^{-at}u(t)\))
• Sinusoids (\(\cos(\omega_0 t)\), \(\sin(\omega_0 t)\))
• Rectangular pulses \(\leftrightarrow\) Sinc functions
• Step function \(u(t)\)
• And others derived from properties.

This slide serves as a summary and a pointer to the reference tables. It’s crucial for students to become familiar with these properties and pairs, as they are the workhorses of Fourier analysis. Encourage students to not just memorize them, but to understand the intuition behind them and how they relate to each other (e.g., duality). The convolution and multiplication properties are arguably the most important for understanding LTI systems and communication principles.

TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM

TABLE 4.2 BASIC FOURIER TRANSFORM PAIRS

Signal	Fourier transform	Fourier series coefficients (if periodic)
\(\sum_{k=-\infty}^{+\infty} a_{k} e^{j k \omega_{0} t}\)	\(2 \pi \sum_{k=-\infty}^{+\infty} a_{k} \delta\left(\omega-k \omega_{0}\right)\)	\(a_{k}\)
\(e^{j \omega_{0} t}\)	\(2 \pi \delta\left(\omega-\omega_{0}\right)\)	\(a_{1}=1\) \(a_{k}=0, \quad\) otherwise
\(\cos \omega_{0} t\)	\(\pi\left[\delta\left(\omega-\omega_{0}\right)+\delta\left(\omega+\omega_{0}\right)\right]\)	\(a_{1}=a_{-1}=\frac{1}{2}\) \(a_{k}=0, \quad\) otherwise
\(\sin \omega_{0} t\)	\(\frac{\pi}{j}\left[\delta\left(\omega-\omega_{0}\right)-\delta\left(\omega+\omega_{0}\right)\right]\)	\(a_{1}=-a_{-1}=\frac{1}{2 j}\) \(a_{k}=0, \quad\) otherwise
\(x(t)=1\)	\(2 \pi \delta(\omega)\)	\(a_{0}=1, \quad a_{k}=0, k \neq 0\) (this is the Fourier series representation for any choice of \(T>0\)
Periodic square wave \(x(t)= \begin{cases}1, & \|t\|<T_{1} \\ 0, & T_{1}<\|t\| \leq \frac{T}{2}\end{cases}\) and \(x(t+T)=x(t)\)	\(\sum_{k=-\infty}^{+\infty} \frac{2 \sin k \omega_{0} T_{1}}{k} \delta\left(\omega-k \omega_{0}\right)\)	\(\frac{\omega_{0} T_{1}}{\pi} \operatorname{sinc}\left(\frac{k \omega_{0} T_{1}}{\pi}\right)=\frac{\sin k \omega_{0} T_{1}}{k \pi}\)
\(\sum_{n=-\infty}^{+\infty} \delta(t-n T)\)	\(\frac{2 \pi}{T} \sum_{k=-\infty}^{+\infty} \delta\left(\omega-\frac{2 \pi k}{T}\right)\)	\(a_{k}=\frac{1}{T}\) for all \(k\)
\(x(t) \begin{cases}1, & \|t\|<T_{1} \\ 0, & \|t\|>T_{1}\end{cases}\)	\(\frac{2 \sin \omega T_{1}}{\omega}\)	-
\(\frac{\sin W t}{\pi t}\)	\(X(j \omega)= \begin{cases}1, & \|\omega\|<W \\ 0, & \|\omega\|>W\end{cases}\)	-
\(\delta(t)\)	1	-
\(u(t)\)	\(\frac{1}{j \omega}+\pi \delta(\omega)\)	-
\(\delta\left(t-t_{0}\right)\)	\(e^{-j \omega t_{0}}\)	-
\(e^{-a t} u(t), \operatorname{Re}\{a\}>0\)	\(\frac{1}{a+j \omega}\)	-
\(t e^{-a t} u(t), \operatorname{Re}\{a\}>0\)	\(\frac{1}{(a+j \omega)^{2}}\)	-
\(\frac{t^{n-1}}{(n-1) !} e^{-a t} u(t)\) \(\operatorname{Re}\{a\}>0\)	\(\frac{1}{(a+j \omega)^{n}}\)	-