Continuous Time Fourier Transform Questions and Answers

Answer: C

The Fourier transform of a rectangular function is a sinc (printed 'Sin C') function.

For a rectangular pulse of height A and width \(\tau\) centred on t = 0:

\[ \begin{aligned} X(f) &= \int_{-\tau/2}^{\tau/2} A e^{-j2\pi f t}\, dt \\ &= A\tau\, \frac{\sin(\pi f \tau)}{\pi f \tau} = A\tau\, \text{sinc}(f\tau) \end{aligned} \]

A sinusoid or cosine in time transforms to impulses in frequency, not the other way round, and an impulse transforms to a constant. So the sinusoidal, cos and impulse options describe transforms of other signals.

Answer: A

The correct matching is a-iii, b-i, c-iv, d-ii.

• The Fourier transform of an impulse \(\delta(t)\) is a constant, 1, at all frequencies (iii).
• A rectangular pulse transforms to a sinc function (i).
• A sine wave of frequency \(f_0\) transforms to a pair of impulses at \(\pm f_0\), its own frequency (iv).
• A DC or constant signal transforms to a single impulse at zero frequency (ii).

Note the duality: an impulse in time is constant in frequency and a constant in time is an impulse in frequency, so the codes that swap (ii) and (iii) are wrong.

Answer: B

The Fourier transform is \(X(jw - w_0)\), the spectrum shifted to \(w_0\).

This is the frequency-shifting (modulation) property. Multiplying a signal by \(e^{jw_0 t}\) shifts its whole spectrum along the frequency axis by \(w_0\):

\[ e^{jw_0 t}\,x(t) \ \leftrightarrow\ X(j(w - w_0)) \]

The paper writes this as \(X(jw - w_0)\). \(X(jw_0)\) is one value of the spectrum, not a shifted spectrum. \(X(jw)\) unchanged ignores the multiplication, and a factor \(e^{-jw_0}\) belongs to the time-shifting property, where a delay multiplies the spectrum by a phase term.

Answer: C

The process is called spectral analysis.

Spectral analysis finds how a signal's power or amplitude is distributed over frequency, using mathematical tools such as the Fourier series, the Fourier transform and, for sampled data, the FFT. The result is the signal's spectrum, which shows its frequency components.

Time domain analysis studies the signal as a function of time, the opposite view. 'Mathematical analysis' is too general to name this process, and 'pseudo analysis' is not a recognised term.