Final answer:
The Fourier transform of the signal x[n] = u[n−2] − u[n−6] is found by expressing the signal as a finite time pulse and computing its transform using the definition of the Fourier transform for discrete-time signals. Ultimately, the transform is expressed as a geometric series that can be simplified further.
Step-by-step explanation:
To compute the Fourier transform of the signal x[n]=u[n−2]−u[n−6], we look at the signal as a difference of two unit step functions. The signal is '1' starting from n=2 and goes back to '0' at n=6. Therefore, the signal can be represented as a pulse of width 4 units. To find the Fourier transform, we need to express this as a summation.
The signal can be written as:
x[n] = Σ_{m=2}^{5} δ[n-m]
Where δ[n] is the Kronecker delta which is 1 when n=0 and 0 otherwise. Applying the definition of the Fourier transform for discrete-time signals:
X(e^{jω}) = Σ_{n=-∞}^{∞} x[n] · e^{-jωn}
Substituting the expression for x[n] we get:
X(e^{jω}) = Σ_{m=2}^{5} e^{-jωm}
The above summation can be solved by recognizing it as a geometric series with a ratio r=e^{-jω}:
X(e^{jω}) = e^{-jω2} · Σ_{k=0}^{3} r^k
Which can be further evaluated if you sum the terms of the geometric series:
X(e^{jω}) = e^{-jω2} · · · [1 + e^{-jω} + e^{-2jω} + e^{-3jω}]
Using the formula for the sum of a geometric series S = a(1 - r^n) / (1 - r) where a is the first term and r is the common ratio:
X(e^{jω}) = e^{-jω2} · (· [1 - e^{-4jω}] / (1 - e^{-jω}))