Why the Fourier transform is needed

The Fourier transform converts a signal from the time domain to the frequency domain: X(ω) = ∫ x(t) e^(-jωt) dt. It is needed for the following reasons.

1. Frequency content: it shows which frequencies are present in a signal and with what amplitude and phase. Many properties, such as bandwidth, cannot be seen from the time-domain waveform.
2. Applies to non-periodic signals: Fourier series only handles periodic signals; the transform handles aperiodic signals such as pulses and transients.
3. Simpler system analysis: convolution in time becomes multiplication in frequency, Y(ω) = X(ω)H(ω), which is far easier than evaluating a convolution integral. The frequency response H(ω) of a system follows directly.
4. Filter design: unwanted frequencies (noise) can be identified and removed with low-pass, high-pass or band-pass filters.
5. Bandwidth and channel design: the spectrum determines the bandwidth needed for transmission and how a channel will distort a signal.
6. Modulation and sampling: modulation is a spectrum shift (multiplication by a carrier), and the sampling theorem is stated in terms of the spectrum.
7. Differential equations: derivatives become multiplication by jω, so differential equations become algebraic.

Example: noise at 50 Hz mixed with a 1 kHz tone is easy to see and remove in the spectrum, but not in the time waveform.