State and prove Parseval's theorem for energy signals

To state and prove Parseval's theorem for energy signals, we first need to define what an energy signal is. An energy signal is a signal that has finite energy, meaning that the integral of its squared magnitude over time is finite.

Now, let's state Parseval's theorem for energy signals:

Parseval's theorem states that the total energy of a signal in the time domain is equal to the total energy of its Fourier transform in the frequency domain.

To prove this theorem, we can start by considering a signal x(t) with Fourier transform X(f). The energy of x(t) can be calculated as the integral of its squared magnitude over time:

E_x = ∫ |x(t)|^2 dt

Similarly, the energy of X(f) can be calculated as the integral of its squared magnitude over frequency:

E_X = ∫ |X(f)|^2 df

Now, we can prove Parseval's theorem by showing that E_x = E_X.

To do this, we can use the Parseval's theorem for Fourier transforms, which states that the integral of the squared magnitude of a signal in the time domain is equal to the integral of the squared magnitude of its Fourier transform in the frequency domain:

∫ |x(t)|^2 dt = ∫ |X(f)|^2 df

Since x(t) is an energy signal, its energy E_x is finite. Therefore, the integral of |x(t)|^2 dt is finite.

Similarly, since X(f) is the Fourier transform of x(t), it is also an energy signal. Therefore, the integral of |X(f)|^2 df is finite.

Since both integrals are finite, we can conclude that E_x = E_X, which proves Parseval's theorem for energy signals.

In summary, Parseval's theorem states that the total energy of a signal in the time domain is equal to the total energy of its Fourier transform in the frequency domain. This theorem can be proven by showing that the integral of the squared magnitude of the signal in the time domain is equal to the integral of the squared magnitude of its Fourier transform in the frequency domain.

To state and prove Parseval's theorem for energy signals, we first need to define what an energy signal is. An energy signal is a signal that has finite energy, meaning that the integral of its squared magnitude over time is finite.

Now, let's state Parseval's theorem for energy signals:

Parseval's theorem states that the total energy of a signal in the time domain is equal to the total energy of its Fourier transform in the frequency domain.

To prove this theorem, we can start by considering a signal x(t) with Fourier transform X(f). The energy of x(t) can be calculated by integrating the squared magnitude of x(t) over time:

E_x = ∫ |x(t)|^2 dt

Similarly, the energy of X(f) can be calculated by integrating the squared magnitude of X(f) over frequency:

E_X = ∫ |X(f)|^2 df

Now, we can prove that E_x = E_X.

Using the inverse Fourier transform, we can express x(t) in terms of its Fourier transform:

x(t) = ∫ X(f) e^(j2πft) df

Substituting this expression into the energy equation for x(t), we get:

E_x = ∫ |∫ X(f) e^(j2πft) df|^2 dt

Expanding the squared magnitude and interchanging the order of integration, we have:

E_x = ∫ ∫ X(f) e^(j2πft) X*(f') e^(-j2πf't) df df' dt

Simplifying the expression, we get:

E_x = ∫ ∫ X(f) X*(f') e^(j2π(f-f')t) df df' dt

Since the exponential term e^(j2π(f-f')t) oscillates rapidly, the integral over time will be zero unless f = f'. Therefore, we can simplify the expression further:

E_x = ∫ |X(f)|^2 df

This is the same expression as the energy of X(f), E_X. Hence, we have proved that E_x = E_X, which is Parseval's theorem for energy signals.

In conclusion, Parseval's theorem states that the total energy of a signal in the time domain is equal to the total energy of its Fourier transform in the frequency domain.