This cheat sheet covers the core Fourier tools used in signals and systems engineering. Students use these tools to represent signals as sums or integrals of sinusoids, which makes many system problems easier to analyze. A compact reference is useful because Fourier formulas have similar forms but different meanings depending on whether the signal is periodic, nonperiodic, continuous time, or discrete time.
Key Facts
- A continuous-time Fourier series represents a periodic signal as x(t) = sum from k = -infinity to infinity of c_k e^(j k omega0 t), where omega0 = 2 pi / T.
- The continuous-time Fourier series coefficient is c_k = (1/T) integral over one period of x(t) e^(-j k omega0 t) dt.
- The continuous-time Fourier transform is X(omega) = integral from -infinity to infinity of x(t) e^(-j omega t) dt.
- The inverse continuous-time Fourier transform is x(t) = (1/2 pi) integral from -infinity to infinity of X(omega) e^(j omega t) d omega.
- Convolution in time becomes multiplication in frequency, so if y(t) = x(t) * h(t), then Y(omega) = X(omega)H(omega).
- Multiplication in time becomes scaled convolution in frequency, so x(t)h(t) transforms to (1/2 pi)[X(omega) * H(omega)].
- A time shift changes phase but not magnitude, so x(t - t0) transforms to e^(-j omega t0)X(omega).
- The sampling frequency must satisfy f_s >= 2 f_max to avoid aliasing for a band-limited signal with highest frequency f_max.
Vocabulary
- Signal
- A signal is a function that carries information, often written as x(t) for continuous time or x[n] for discrete time.
- Fourier Series
- A Fourier series represents a periodic signal as a weighted sum of complex sinusoids at integer multiples of a fundamental frequency.
- Fourier Transform
- A Fourier transform represents a nonperiodic signal by describing how much of each continuous frequency is present.
- Spectrum
- A spectrum is the frequency-domain description of a signal, usually showing magnitude and phase versus frequency.
- Convolution
- Convolution combines an input signal with a system impulse response to produce the output of a linear time-invariant system.
- Aliasing
- Aliasing is distortion that occurs when sampling is too slow, causing high frequencies to appear as lower frequencies.
Common Mistakes to Avoid
- Mixing angular frequency omega and ordinary frequency f is wrong because omega = 2 pi f, so missing the factor of 2 pi changes formulas and units.
- Forgetting the 1/T factor in Fourier series coefficients is wrong because c_k must measure the average contribution of one harmonic over a full period.
- Treating convolution as ordinary multiplication in the time domain is wrong because convolution integrates shifted overlap, while multiplication only combines point-by-point values.
- Ignoring phase is wrong because two signals can have the same magnitude spectrum but different time-domain shapes due to different phase relationships.
- Using a sampling rate below 2 f_max is wrong for a band-limited signal because frequencies above half the sampling rate fold into lower frequencies and create aliasing.
Practice Questions
- 1 A periodic signal has period T = 0.004 s. Find its fundamental frequency f0 and angular frequency omega0.
- 2 If x(t) has Fourier transform X(omega), what is the Fourier transform of x(t - 3)?
- 3 A sensor signal contains frequencies up to 4.5 kHz. What minimum sampling frequency is needed to avoid aliasing?
- 4 Explain why analyzing an LTI system in the frequency domain can be easier than analyzing it directly in the time domain.