ELEC2134 Chap.7 Fourier Transform Properties
Fourier Transform Properties
Transform properties let you reuse a known pair after a signal is delayed, scaled, modulated, differentiated or convolved. The main challenge is bookkeeping: time scaling changes both the frequency argument and amplitude, while time shift changes phase without changing magnitude.
This chapter chains operations in a declared order and uses area, bandwidth and phase predictions to catch reversed arguments or signs.
Nested signal arguments should be rewritten before a rule is selected. Separating a scale from an effective delay shows which operation occurs first and where each factor belongs.
One line per property creates an audit trail: the frequency argument, amplitude multiplier and phase factor can each be traced to a named time-domain change rather than appearing by guesswork.
Qualitative predictions make the chain safer. Compression in time must broaden the spectrum, delay must preserve magnitude, cosine modulation must create two shifted copies, and differentiation must remove a finite DC component.
Convolution is the bridge to systems because a time-domain overlap integral becomes an ordinary spectral product. If the final formula contradicts one of these predictions, repair the earliest property line.
What this chapter covers
- 01
Time shift and linear phase
- 02
Time scaling and inverse bandwidth change
- 03
Frequency shift through complex modulation
- 04
Cosine modulation and two shifted copies
- 05
Differentiation as multiplication by j omega
- 06
Convolution as frequency-domain multiplication
Worked example · free
Scale and delay a known transform pair
- +1Define z(t) = x(2t), so y(t) = 3z(t - 1). This exposes scaling before delay.
- +2Time scaling gives Z(omega) = one half X(omega/2) = 1/(4 + j omega).
- +2The one-second delay multiplies by exp(-j omega), and the outer amplitude multiplies by 3.
- +1Thus Y(omega) = 3 exp(-j omega)/(4 + j omega). Its zero-frequency value is 3/4, matching time-domain area.
Key terms
- Time shift
- A delay or advance that introduces linear spectral phase without changing magnitude.
- Time scaling
- Compression or expansion in time accompanied by inverse frequency-axis scaling and an amplitude factor.
- Frequency shift
- Translation of a spectrum caused by multiplication with a complex exponential in time.
- Differentiation property
- The transform rule mapping a time derivative to multiplication by j omega.
- Modulation
- Multiplication by a carrier that moves baseband spectral content to other frequencies.
Fourier Transform Properties FAQ
Why does a time delay not change transform magnitude?
The delay factor exp(-j omega t0) has magnitude one at every frequency. It rotates each spectral component by a frequency-dependent phase but cannot scale its magnitude. A proposed delayed spectrum with a different envelope therefore violates the shift property, even if its phase was not requested.
What is the most common time-scaling error?
Students often change the argument to X(omega/a) but forget the factor 1/|a|, or use X(a omega) instead. Remember the width tradeoff: compressing time by |a| widens frequency by |a|. The amplitude factor then preserves the transform's area relationship under the chosen convention.
How does cosine modulation differ from complex modulation?
Multiplication by exp(j omega0 t) creates one shifted copy X(omega - omega0). Multiplication by cos(omega0 t) uses two exponentials, so it creates half-amplitude copies at both positive and negative shifts. For a real signal, those copies combine consistently with conjugate symmetry.
What does differentiation do to DC content?
Multiplication by j omega forces the transformed derivative to zero at omega equals zero when the original transform is finite there. This matches the idea that differentiating removes a constant component. If the time signal has jumps, include the associated impulse contributions before using an ordinary derivative expression.
How can I reverse a property chain?
Start from the most recognisable spectral parent and remove one feature at a time. A phase ramp suggests a time shift, translated copies suggest modulation and a product may suggest time convolution. After proposing the time signal, transform its structure forward again to verify every scale, shift and factor.
Exam move
Build a one-page property table and annotate what each operation does to magnitude, phase and bandwidth. In multi-operation problems, rewrite the argument into a standard form and apply one rule per line. Predict the final shape before simplifying. Use X(0), magnitude preservation under delay and conjugate symmetry as independent checks, and practise reversing a property as well as applying it forward.
Drill near-miss pairs such as X(omega/a) versus X(a omega), and one complex shift versus two cosine-generated shifts. After each answer, point to the exact factor created by every operation.
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