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The Fourier Transform X(f) = ∫ x(t)e-j2πft dt converts aperiodic signals from the time domain to the frequency domain. Key properties include linearity, time shift, frequency shift, scaling, and duality, which simplify analysis of complex signals and systems.
You already know from Fourier Series that a periodic signal can be decomposed into a sum of sinusoids at harmonically related frequencies. But what happens to a signal that never repeats — a single pulse, a decaying exponential, a one-time radar ping? The Fourier Series cannot handle this directly, because it assumes periodicity. The Fourier Transform is the generalization that handles arbitrary aperiodic signals.
The definition is X(f) = ∫_{-∞}^{∞} x(t) e-j2πft dt. Think of it this way: for each frequency f, you multiply the signal x(t) by a complex sinusoid oscillating at that frequency and integrate (average) the result over all time. If x(t) contains a lot of energy at frequency f, the product x(t)e-j2πft adds up constructively and X(f) is large there. If x(t) has no component at f, the oscillations cancel and X(f) is near zero. The result X(f) is generally complex — its magnitude |X(f)| is the amplitude spectrum (how much of each frequency is present) and its angle ∠X(f) is the phase spectrum (the timing relationship between frequency components).
The properties of the Fourier Transform are where its power comes from. Linearity means transforms of sums are sums of transforms — you can analyze components separately. The time-shift property says delaying a signal in time multiplies its transform by a complex exponential e-j2πft₀: the amplitude spectrum is unchanged, only the phases shift. The frequency-shift (modulation) property is the dual: multiplying x(t) by ej2πf₀t shifts the spectrum to be centered at f₀ — exactly what a radio transmitter does when it modulates a signal onto a carrier. The scaling property says compressing a signal in time stretches its spectrum in frequency (and vice versa): a narrow pulse has a wide bandwidth.
The duality property is particularly elegant: if X(f) is the transform of x(t), then x(f) is the transform of X(-t). The transform and its inverse have nearly the same mathematical form. This duality means every time-domain result has a corresponding frequency-domain result — if you know what a rectangular pulse looks like in frequency (a sinc function), duality immediately tells you what a sinc-shaped signal looks like in time (a rectangular spectrum).
The Fourier Transform connects to Fourier Series in a precise way: as the period of a periodic signal is taken to infinity, the discrete harmonic frequencies of the series merge into a continuous frequency axis and the sum becomes an integral. This is why the Fourier Transform is often called the "aperiodic limit" of the Fourier Series — they describe the same underlying phenomenon of frequency decomposition, just for different signal classes.