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.
Understanding Signals & Systems Fourier Reference
A periodic waveform has a fundamental frequency set by how long one cycle lasts. Its spectrum contains harmonics at whole-number multiples of that fundamental. The coefficient for each harmonic carries two pieces of information.
Its magnitude tells how strongly that harmonic is present. Its phase tells where that sinusoidal component sits in time relative to a reference. Negative frequencies can seem strange at first.
They come from using rotating complex exponentials to describe sinusoids. For real physical signals, the positive and negative parts have a matching symmetry. This symmetry is a useful error check when drawing or calculating a spectrum.
A Fourier transform describes how frequency content is spread rather than restricted to harmonic lines. A short pulse has a broad spectrum because it changes quickly and exists for only a short time. A long, steady tone has a narrow spectrum near one frequency.
This is an important tradeoff in measurements. Recording a signal for a longer time gives finer frequency detail. Cutting a signal into a short time window makes its spectrum wider and can create leakage, where energy appears in nearby frequency bins.
Students should keep track of whether frequency is measured in hertz or in radians per second. The conversion uses two pi radians for each cycle, and mixing these units causes many missing-factor errors.
Convolution describes the output of a linear time-invariant system from its input and impulse response. The impulse response is the system's reaction to a very brief input. It acts like a fingerprint of the system.
A room creates echoes, an electronic circuit smooths voltage changes, and a camera lens blurs an image. Each can be modeled with an impulse response. In the frequency view, the system has a frequency response.
Frequencies where its magnitude is small are reduced. Frequencies where its magnitude is large pass more strongly. Phase matters too.
A filter can preserve the magnitude of a signal while shifting different frequency components by different amounts. This can change the shape of pulses and data signals.
Modulation moves a signal's spectrum to a new frequency range. Radio transmitters use this idea to place many messages in separate bands so receivers can select one station. Sampling creates a discrete record of a continuous signal, as in digital audio and sensor logging.
If the sample rate is too low, high-frequency content folds into lower frequencies and becomes false content. This aliasing cannot usually be repaired after recording. An anti-alias filter removes unwanted high frequencies before sampling.
When studying worked problems, first label the signal type, its time units, and its frequency units. Then identify whether the operation is a shift, scaling, multiplication, convolution, modulation, or sampling. This classification often tells you the correct Fourier property before any calculation begins.