Digital Signal Processing Basics
Which of the following signals takes the same value as the discrete-time signal x[n] = cos(1.75πn) for every integer n?
- Acos(0.25πn)
- Bcos(0.75πn)
- Ccos(0.875πn)
Show answer and explanation
Answer: A. cos(0.25πn)
Because n is an integer, a discrete-time sinusoid gives exactly the same sequence of values when its angular frequency is shifted by 2π. cos(1.75πn) = cos(1.75πn − 2πn) = cos(−0.25πn), and since cos is an even function this equals cos(0.25πn). This is aliasing seen from the digital side: a frequency above ω = π cannot be told apart from the lower frequency 2π − ω. In terms of the sampling frequency, a sine wave at 0.875fs gives the same samples as one at 0.125fs.
- A. Correct. 1.75π − 2π = −0.25π, and cos is even, so the values are the same.
- B. This subtracts π from 1.75π. A shift of π gives cos(1.75πn) = (−1)ⁿcos(0.75πn), so the odd-numbered samples change sign.
- C. This is 1.75π divided by 2. Halving the angular frequency does not give the same sequence of values.