Time Scaling property of the Fourier Transform

Electrical Circuit Analysis > Fourier Transform > Properties of the Fourier Transform

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Introduction

The Time Scaling Property of the Fourier Transform is an important concept in signal processing and electrical engineering. It explains how the frequency spectrum of a signal changes when the signal is compressed or expanded in the time domain. This property is widely used in communication systems, audio processing, and control systems.

When a signal is compressed in time, its frequency components spread out, and when it is expanded, the frequency components become more concentrated. Understanding this relationship helps engineers analyze and manipulate signals effectively.

Definition of Time Scaling Property

If a time-domain signal $ x(t)$ has a Fourier Transform $X(f)$, then scaling the time variable by a constant $ a $ gives: $$ x(at) \quad \Longleftrightarrow \quad \frac{1}{|a|} X\left(\frac{f}{a}\right) $$

Where:

a = scaling factor
x(t) = original signal
X(f) = Fourier Transform of the signal

Key Observations

1. Time Compression (|a| > 1)

If the value of $ a $ is greater than 1, the signal is compressed in time.

Effects:

The signal becomes narrower
The frequency spectrum becomes wider
Amplitude decreases by factor $1/|a| $

2. Time Expansion (|a| < 1)

If the value of $a$ is less than 1, the signal is expanded.

Effects:

The signal becomes wider
The frequency spectrum becomes narrower
Amplitude increases accordingly

3. Negative Scaling (a < 0)

If the scaling factor is negative, the signal is reversed in time.

Effects:

Signal flips along time axis
Frequency spectrum also reflects

If $ F(\omega)=\mathcal{F}[f(t) ]$, then

$$\mathcal{F}[f(a t)]=\frac{1}{|a|} F\left(\frac{\omega}{a}\right) \tag{1}$$

where $ a $ is a constant. Equation (1) shows that time expansion $ (|a|>1) $ corresponds to frequency compression, or conversely, time compression $ (|a|<1) $ implies frequency expansion. The proof of the time-scaling property proceeds as follows.

$$\mathcal{F}[f(a t)]=\int_{-\infty}^{\infty} f(a t) e^{-j \omega t} d t \tag{2}$$

If we let $ x=a t $, so that $ d x=a d t $, then

$$\mathcal{F}[f(a t)]=\int_{-\infty}^{\infty} f(x) e^{-j \omega x / a} \frac{d x}{a}=\frac{1}{a} F\left(\frac{\omega}{a}\right) \tag{3}$$

For example, for the rectangular pulse $ p(t) $, $$\mathcal{F}[p(t)]=A \tau \operatorname{sinc} \frac{\omega \tau}{2} \tag{4.1}$$ Using Eq. (1),

$$\mathcal{F}[p(2 t)]=\frac{A \tau}{2} \operatorname{sinc} \frac{\omega \tau}{4} \tag{4.2}$$

It may be helpful to plot $ p(t) $ and $ p(2 t) $ and their Fourier transforms. Since

$$ p(t)= \begin{cases} A, & -\frac{\tau}{2} < t < \frac{\tau}{2} \\ 0, & \text{otherwise} \end{cases} $$

then replacing every $ t $ with $ 2 t $ gives

$$ p(2t) = \begin{cases} A, & -\frac{\tau}{4} < t < \frac{\tau}{4} \\ 0, & \text{otherwise} \end{cases} $$

showing that $ p(2 t) $ is time compressed, as shown in Fig. 1(b). To plot both Fourier transforms in Eq. (4), we recall that the sinc function has zeros when its argument is $ n \pi $, where $ n $ is an integer.

Fig. 1: The effect of time scaling: (a) transform of the pulse, (b) time compression of the pulse causes frequency expansion.

Hence, for the transform of $ p(t) $ in Eq. (4.1),

$$ \omega \tau / 2=2 \pi f \tau / 2=n \pi \rightarrow f=n / \tau $$

and for the transform of $ p(2 t) $ in Eq. (4.2),

$$ \omega \tau / 4=2 \pi f \tau / 4= $ $ n \pi \rightarrow f=2 n / \tau $$

The plots of the Fourier transforms are shown in Fig. 1, which shows that time compression corresponds with frequency expansion. We should expect this intuitively, because when the signal is squashed in time, we expect it to change more rapidly, thereby causing higher-frequency components to exist.

Physical Interpretation

The time scaling property shows an inverse relationship between time and frequency.

Short signals contain high-frequency components
Long signals contain low-frequency components

This means that reducing the duration of a signal increases its bandwidth, while increasing its duration reduces its bandwidth.

Mathematical Insight

The scaling factor $ \frac{1}{|a|} $ ensures that the energy of the signal remains constant after scaling.

This is important because physical systems must conserve energy, and the Fourier Transform maintains this property.

Example of Time Scaling

Example: If the Fourier Transform of a signal $x(t)$ is $X(f) $, find the transform of $ x(2t) $.

Solution: Using the time scaling property: $$ x(2t) \quad \Longleftrightarrow \quad \frac{1}{2} X\left(\frac{f}{2}\right) $$

This means:

The signal is compressed in time
The frequency spectrum expands
Amplitude becomes half

Graphical Understanding

Consider a signal plotted in time domain:

Compression makes the signal narrow and tall
Expansion makes the signal wide and short
The frequency plot behaves in the opposite way

This inverse relationship is fundamental in signal processing.

Applications of Time Scaling

Audio Processing: Speeding up or slowing down audio signals
Communication Systems: Bandwidth control
Image Processing: Scaling images in frequency domain
Radar Systems: Pulse compression techniques

Important Points

Time and frequency are inversely related
Scaling in time affects bandwidth
Energy remains conserved
Negative scaling reverses the signal

Conclusion

The Time Scaling Property of the Fourier Transform is essential for understanding how signals behave under compression and expansion. It highlights the inverse relationship between time and frequency and plays a vital role in modern engineering applications such as communication, signal processing, and control systems.

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