Consider two continuous time signals $x(t)$ and $y(t)$ as shown below 
If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is _________
To find the Fourier transform of \( y(t) \) in terms of \( X(f) \), we first establish a mathematical relationship between the two signals \( x(t) \) and \( y(t) \) using time-domain transformations.
1. Relationship between \( x(t) \) and \( y(t) \)
- Amplitude Inversion: Signal \( x(t) \) has a negative peak (\(-1\)), while \( y(t) \) has a positive peak (\(1\)). This indicates a reflection across the time axis: \( -x(t) \).
- Time Scaling: The base width of \( x(t) \) is \( 1 \) unit (from \(t = -1\) to \( t = 0 \)). The base width of \( y(t) \) is \( 4 \) units (from \(t = -2\) to \( t = 2 \)). This is a time expansion by a factor of \( 4 \), achieved by replacing \( t \) with \( t/4 \).
- Time Shifting:
- The peak of \( x(t) \) is at \( t = -0.5 \).
- When we expand the time scale by \( 4 \), the peak position moves to \( t = -0.5 \times 4 = -2 \).
- To align this expanded peak with the peak of \( y(t) \) at \( t = 0 \), we must shift the signal to the right by \( 2 \) units. Replacing \( t \) with \( t - 2 \) in the scaled expression gives: $$ y(t) = -x\left( \frac{t - 2}{4} \right) $$
Verification:
- At \( t = 0 \): \( y(0) = -x(-2/4) = -x(-0.5) = -(-1) = 1 \). (Correct)
- At \( t = 2 \): \( y(2) = -x(0) = 0 \). (Correct)
- At \( t = -2 \): \( y(-2) = -x(-1) = 0 \). (Correct)
2. Applying Fourier Transform Properties
We use the following standard properties of the Fourier Transform:
- Time Scaling: If \( \mathcal{F}\{x(t)\} = X(f) \), then \( \mathcal{F}\{x(at)\} = \frac{1}{|a|} X\left( \frac{f}{a} \right) \).
- Time Shifting: If \( \mathcal{F}\{x(t)\} = X(f) \), then \( \mathcal{F}\{x(t - t_0)\} = X(f) e^{-j 2\pi f t_0} \).
Starting with \( y(t) = -x\left( \frac{1}{4}t - \frac{1}{2} \right) \), let \( g(t) = x(t/4) \). Using the scaling property with \( a = 1/4 \):
$$ G(f) = \frac{1}{1/4} X\left( \frac{f}{1/4} \right) = 4 X(4f) $$
Now, rewrite \( y(t) \) as \( y(t) = -g(t - 2) \). Using the shifting property with \( t_0 = 2 \):
$$ Y(f) = -G(f) e^{-j 2\pi f (2)} $$ $$ Y(f) = -4 X(4f) e^{-j 4\pi f} $$
Conclusion
Comparing our derived result with the provided options, the correct expression for the Fourier transform of \( y(t) \) is:
$$ -4X(4f)e^{-j 4\pi f} $$
Important Questions from Properties of Fourier Transform
Which one of the following is the Fourier transform of the signal given in Fig. II, if the Fourier transform of the signal in Fig. I is \(=2\dfrac{\sin\omega T_1}{\omega}\) ?
The Hilbert transform of cos ω1t + sin ω2t is
Let $X(\omega)$ be the Fourier transform of the signal
$x(t) = e^{-t^4} \cos t$, $-\infty < t < \infty$.
The value of the derivative of $X(\omega)$ at $\omega=0$ is ________ (rounded off to 1 decimal place).- Let $f(t)$ and $g(t)$ represent continuous-time real-valued signals. If $h(t)$ denotes cross-correlation between $f(t)$ and $g(-t)$, its continuous-time Fourier transform $H(j\omega)$ equals
Note: $F(j\omega)$ and $G(j\omega)$ denote the continuous-time Fourier transforms of $f(t)$ and $g(t)$, respectively. Consider a continuous-time, real-valued signal $f(t)$ whose Fourier transform $F(\omega) = \int_{-\infty}^{\infty} f(t)exp(-j\omega t)dt$ exists.
Which one of the following statements is always TRUE?
