Final answer:
The signal X3[n]= eᵗ⁷πⁿ is periodic with a fundamental period of 2 since eᵗ⁷πN must be 1, which happens when N is 2.
Step-by-step explanation:
The student asks whether the signal X₃[n]= eᵗ⁷πⁿ is periodic and, if so, what its fundamental period is. A signal is periodic if it repeats itself at regular intervals. For a discrete-time complex exponential signal like this, we look for a period N such that eᵗ⁷πⁿ = eᵗ⁷π(n+N) for all n. Solving this equation:
eᵗ⁷πⁿ = eᵗ⁷π(n)eᵗ⁷πN
eᵗ⁷πN must be equal to 1 for the signal to be periodic.
Since eᵗ⁷πN = cos(7πN) + i sin(7πN), we need cos(7πN) to be 1 and sin(7πN) to be 0.
These conditions are satisfied when 7πN is an integer multiple of 2π, or N is an integer multiple of 2.
Thus, X₃[n] is periodic and its fundamental period N is the smallest positive integer that satisfies these conditions, which is 2. Therefore, the fundamental period of the signal X₃[n] is 2.
To determine whether the signal X₃[n] = e^(i7πn) is periodic, we need to check if there is a fundamental period for the signal. A signal is periodic if it repeats itself after a certain interval of time, called the period. In this case, the signal is in the form of e^(iθ), where θ is a constant value. For the signal to be periodic, θ has to be a multiple of 2π.
Let's check if θ = 7π is a multiple of 2π:
7π = 3.5 * 2π
Since 7π can be expressed as a multiple of 2π, the signal X₃[n] is periodic. The fundamental period of the signal is the smallest positive value of n for which X₃[n] repeats itself.