4.1 Linearity, Time-Shifting, and Frequency-Shifting

Linearity of the Fourier Transform

The linearity property lets you break a complex signal into simpler pieces, transform each piece separately, and then combine the results. This is the reason Fourier analysis works so well for real systems: most systems you'll encounter are linear, so you can analyze them one component at a time.

Linearity Property and Its Implications

If $x(t)$ and $y(t)$ are two signals with Fourier transforms $X(\omega)$ and $Y(\omega)$, and $a$ and $b$ are constants, then:

$\mathcal{F}\{a\,x(t) + b\,y(t)\} = a\,X(\omega) + b\,Y(\omega)$

In plain terms: scaling and addition pass straight through the Fourier transform. You can transform first and combine later, or combine first and transform later. The result is the same.

This matters most when working with linear time-invariant (LTI) systems. Because LTI systems obey superposition, you can:

1. Decompose a complicated input into simpler component signals.
2. Take the Fourier transform of each component individually.
3. Multiply each transform by the system's transfer function $H(\omega)$ to get the output spectrum.
4. Sum the results (by linearity) and inverse-transform to get the time-domain output.

Applications and Importance of Linearity

Superposition is the practical payoff of linearity. The response of a linear system to a sum of inputs equals the sum of the responses to each input on its own. Telecommunications systems, for example, routinely process many signals at once and rely on this fact.

Linearity also connects directly to the convolution theorem: convolution of two signals in the time domain becomes multiplication of their Fourier transforms in the frequency domain.

$\mathcal{F}\{x(t) * h(t)\} = X(\omega) \cdot H(\omega)$

This turns a potentially painful integral (convolution) into straightforward multiplication, which is one of the biggest practical advantages of working in the frequency domain.

Time and Frequency Shifting

Time-Shifting Property

If $x(t)$ has Fourier transform $X(\omega)$, then delaying the signal by $t_0$ seconds gives:

$\mathcal{F}\{x(t - t_0)\} = e^{-j\omega t_0} \cdot X(\omega)$

The key insight: a time delay does not change the magnitude spectrum $|X(\omega)|$. It only introduces a linear phase shift of $-\omega t_0$. The phase shift grows linearly with frequency $\omega$ and is proportional to the delay $t_0$.

This makes intuitive sense. Shifting a signal in time doesn't add or remove any frequency content; it just changes when those frequency components arrive. That "when" information lives entirely in the phase.

Practical examples:

• An echo in an audio signal is a delayed copy of the original. In the frequency domain, the echo has the same magnitude spectrum but a different phase.
• Frame delays in video processing work the same way: the content is identical, just shifted in time.

Frequency-Shifting Property

If $x(t)$ has Fourier transform $X(\omega)$, then multiplying by a complex exponential at frequency $\omega_0$ gives:

$\mathcal{F}\{x(t)\,e^{j\omega_0 t}\} = X(\omega - \omega_0)$

This shifts the entire spectrum to the right by $\omega_0$. Multiplying by $e^{-j\omega_0 t}$ would shift it to the left by $\omega_0$.

Notice the duality with time-shifting: a shift in one domain corresponds to multiplication by a complex exponential in the other domain.

Practical examples:

• Amplitude modulation (AM): Multiplying a baseband signal by $\cos(\omega_c t)$ shifts its spectrum up to a carrier frequency $\omega_c$ (and creates a mirror image at $-\omega_c$), since $\cos(\omega_c t) = \frac{1}{2}(e^{j\omega_c t} + e^{-j\omega_c t})$.
• Frequency-division multiplexing (FDM): Different signals are shifted to different carrier frequencies so they can share the same channel without interfering.

Fourier Transform of Shifted Signals

Applying Time and Frequency Shifting Together

When a signal undergoes both a time shift and a frequency shift, you apply both properties. The order matters for getting the phase factor right.

For $x(t - t_0)\,e^{j\omega_0 t}$, the Fourier transform is:

$e^{-j(\omega - \omega_0)t_0} \cdot X(\omega - \omega_0)$

Here's a step-by-step approach for finding the transform of a shifted signal:

1. Start with the known transform $X(\omega)$ of the unshifted signal $x(t)$.

2. Apply the frequency shift if the signal is multiplied by $e^{j\omega_0 t}$: replace $\omega$ with $\omega - \omega_0$ in $X(\omega)$.

3. Apply the time shift if the signal is delayed by $t_0$: multiply the result by $e^{-j\omega t_0}$.

4. Simplify the combined expression.

Analyzing Shifted Signals in Practice

These properties show up constantly in real systems:

• Radar ranging: The received radar pulse is a time-delayed version of the transmitted pulse. Measuring the phase shift across frequencies lets you extract the round-trip delay $t_0$, which gives target range as $R = \frac{c \cdot t_0}{2}$.
• Doppler radar: A moving target also frequency-shifts the return signal. The Doppler shift $\omega_d$ is proportional to the target's radial velocity, so you can extract both range (from time shift) and speed (from frequency shift) simultaneously.
• Digital communications: Modulation schemes like QAM and OFDM use frequency shifting to place data on specific carrier frequencies. Understanding how shifts affect the spectrum is essential for designing receivers that correctly demodulate these signals.
• Matched filtering and pulse compression rely on knowing exactly how time and frequency shifts transform the signal spectrum, so the receiver can optimally detect delayed, Doppler-shifted returns.