Questions: Magnitude and Phase Spectrum Representation

Question 1

You have a speech recording X(f). You replace all phase values φ(f) with random numbers while keeping |X(f)| exactly unchanged. What will the result sound like when played back?

• A. Identical to the original — phase is a mathematical artifact with no perceptual effect
• B. Slightly noisier but still intelligible, since the frequency content is preserved
• C. Like broadband noise — all the frequency amplitudes are correct but intelligibility is destroyed
• D. Louder, because randomizing phase causes constructive interference at more time points

Answer: (C)

Phase encodes the temporal structure of the signal — when each frequency component starts relative to t = 0 and how the components align with each other. Scrambling phase while preserving magnitude destroys the precise timing relationships that make speech intelligible, music coherent, or any waveform reconstructible. The resulting signal has the correct 'color' (same frequency power distribution) but sounds like noise because no useful pattern exists in the time domain. This is a famous demonstration of how much information is carried by phase, contradicting the common misconception that magnitude alone captures 'what the signal is.'

Question 2

A signal x(t) is delayed by 3 seconds to produce y(t) = x(t − 3). How does this change the magnitude and phase spectra?

• A. Both magnitude and phase are unchanged — a time delay is a passive operation
• B. The magnitude spectrum is unchanged; the phase spectrum shifts by −2πf × 3 (a linear function of frequency)
• C. The magnitude spectrum shifts left by 3 Hz; the phase is unchanged
• D. Both magnitude and phase are multiplied by e^(−j6π)

Answer: (B)

This is the time-shift property of the Fourier transform: Y(f) = e^(−j2πf·t₀) · X(f). Multiplying by e^(−j2πft₀) is a pure rotation in the complex plane at each frequency — it changes the phase by −2πft₀ but has unit magnitude, leaving |Y(f)| = |X(f)| unchanged. The phase shift is linear in frequency, which is the hallmark of a pure time delay. This property is critical in practice: if two sensors receive the same signal at slightly different times, you can identify the time delay by comparing their phase spectra and measuring the slope of the phase difference.

Question 3

For a real-valued signal, the magnitude spectrum is an even function of frequency and the phase spectrum is an odd function.

• T. True
• F. False

Answer: True

This follows from the Hermitian symmetry of the Fourier transform of real signals: X(−f) = X*(f) (complex conjugate). Taking magnitudes: |X(−f)| = |X*(f)| = |X(f)| — even symmetry. Taking arguments: φ(−f) = arg(X*(f)) = −arg(X(f)) = −φ(f) — odd symmetry. This symmetry means that for real signals, the positive-frequency half of the spectrum completely determines the signal; the negative-frequency half contains redundant information. In practice, it means you only need to plot and work with positive frequencies when analyzing real signals.

Question 4

The magnitude spectrum of a signal contains most of the information needed to reconstruct the original signal, since it shows the amplitude of nearly every frequency component present.

• T. True
• F. False

Answer: False

Both magnitude AND phase are required for reconstruction. The magnitude spectrum tells you how much of each frequency is present, but not when or in what phase. Two completely different signals can have identical magnitude spectra but entirely different phase spectra — and they will look and sound nothing alike. A pure cosine and a pure sine at the same frequency have the same magnitude spectrum but differ by a 90° phase shift. A signal and its mirror image can have the same magnitude spectrum but opposite phase. Phase is not redundant — it encodes the temporal structure that the magnitude spectrum discards.

Question 5

Explain in your own words why both the magnitude and phase spectrum are necessary to fully describe a signal, using a concrete analogy or example.

Model answer: Magnitude tells you 'how much of each frequency is present' — the loudness of each pitch in a piece of music, say. Phase tells you 'when each frequency component starts, relative to the others' — how the pitches are synchronized in time. Knowing every pitch and its volume (magnitude) is not enough to reconstruct the music; you also need to know the timing (phase) of each note. Scramble the timing while keeping the pitches and volumes unchanged, and you get noise instead of music. Both pieces of information together uniquely determine the signal.

A more mathematical way to see this: the Fourier transform maps each signal to a unique complex-valued function X(f). Two different signals correspond to two different complex functions. If you only record the magnitude |X(f)| and discard the phase, you discard the imaginary part of X(f) after conversion to polar form — an entire degree of freedom per frequency. Without phase, you cannot perform the inverse Fourier transform and recover x(t). The magnitude spectrum alone corresponds to many different possible signals, not a unique one.