Test: Discrete Time Signal Analysis - 1

Explanation: Here, the frequency F0 of a continuous time signal is divided into 2π/N intervals.
So, the Fourier series representation of a discrete time signal with period N is given as

where c_k is the Fourier series coefficient

Explanation: We know that, the Fourier series representation of a discrete signal x(n) is given as

Explanation: We know that,
In the above equation, ck represents the amplitude and ej2πkn/N represents the phase associated with the frequency component of DTFS.

Explanation: For ω₀=√2π, we have f₀=1/√2. Since f₀ is not a rational number, the signal is not periodic. Consequently, this signal cannot be expanded in a Fourier series.

Explanation: In this case, f0=1/6 and hence x(n) is periodic with fundamental period N=6.
Given signal is x(n)= cosπn/3=cos2πn/6=1/2 e^(j2πn/6)+1/2 e^(-j2πn/6)
We know that -2π/6=2π-2π/6=10π/6=5(2π/6)

So, we get c1=c2=c3=c4=0 and c1=c5=1/2.

Explanation: Here, the frequency F0 of a continuous time signal is divided into 2π/N intervals.
So, the Fourier series representation of a discrete time signal with period N is given as

where c_k is the Fourier series coefficient

Explanation: Let us consider a discrete time periodic signal x(n) with period N.
The average power of that signal is given as

Explanation: We know that

Explanation: If we consider a signal x(n) which is discrete in nature and has finite energy, then the Fourier transform of that signal is given as

Explanation: Let X(ω) be the Fourier transform of a discrete time signal x(n) which is given as

Now
So, the Fourier transform of a discrete time finite energy signal is periodic with period 2π.

Explanation: We know that the Fourier transform of the discrete time signal x(n) is

The above equation is known as synthesis equation or inverse transform equation.

Explanation: We know that,

At n = 0 

Therefore, the value of the signal x(n) at n=0 is ω_c/π.

Explanation: We know that,

Explanation: We note that there is a significant oscillatory overshoot at ω=ωc, independent of the value of N. As N increases, the oscillations become more rapid, but the size of the ripple remains the same. One can show that as N→∞, the oscillations converge to the point of the discontinuity at ω=ωc. The oscillatory behavior of the approximation of XN(ω) to the function X(ω) at a point of discontinuity of X(ω) is known as Gibbs phenomenon.

Explanation: We know that,