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Aliasing; Nyquist of 5cos1500πt − …

Short answer

Aliasing is distortion when sampling below the Nyquist rate causes spectral overlap that folds high frequencies into lower ones. For x(t)=5cos(1500πt)−2sin(2000πt)+3sin(1850πt), frequencies are 750, 1000, and 925 Hz; maximum is 1000 Hz, so Nyquist rate is 2000 Hz.

What do you mean by aliasing? Compute Nyquist rate for x(t) = 5cos(1500πt) − 2sin(2000πt) + 3sin(1850πt).

Aliasing

Aliasing is the distortion that occurs when a signal is sampled at a rate lower than its Nyquist rate (fs<2fMf_s < 2f_M). The shifted copies of the spectrum in the sampled signal overlap, so a high-frequency component "folds back" and appears as a lower frequency (its alias). After overlap the original spectrum cannot be separated by any filter, so the signal cannot be recovered exactly.

Example: a 700 Hz tone sampled at fs=1000f_s = 1000 Hz gives the same samples as a 1000−700=3001000-700 = 300 Hz tone.

   overlap (aliasing) when ws < 2wM
     ___       ___       ___
    /   \     /   \     /   \
   /     \   /     \   /     \
--/-------\-/-------\-/-------\--> w
       -ws  \ /  0   \ /  ws
             X        X   <- overlapped region

Nyquist rate of x(t)=5cos⁡(1500πt)−2sin⁡(2000πt)+3sin⁡(1850πt)x(t) = 5\cos(1500\pi t) - 2\sin(2000\pi t) + 3\sin(1850\pi t)

Termω\omega (rad/s)f=ω/2πf = \omega/2\pi (Hz)
5cos⁡(1500πt)5\cos(1500\pi t)1500π1500\pi750750
−2sin⁡(2000πt)-2\sin(2000\pi t)2000π2000\pi10001000
3sin⁡(1850πt)3\sin(1850\pi t)1850π1850\pi925925

The maximum frequency is fM=1000f_M = 1000 Hz.

fN=2fM=2×1000=2000 Hzf_N = 2 f_M = 2 \times 1000 = 2000\ \text{Hz}

Answer: Nyquist rate = 2000 Hz (2000 samples/s, i.e. ωN=4000π\omega_N = 4000\pi rad/s).

✓ Answer checked by Claude Opus 5.5 (high effort)

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