When a time-domain signal is converted into its Fourier representation, which of the following is/are conserved? I. Energy II. Power
I and II
Fourier Representation and Signal Conservation
When a signal undergoes a transformation from the time domain to its Fourier representation, it's important to understand what properties of the signal are preserved. The question specifically asks about the conservation of energy and power during this conversion. This concept is fundamental in signal processing and is directly related to Parseval's theorem.
Energy Conservation in Fourier Representation
The conservation of energy between the time domain and the frequency (Fourier) domain is a key principle, often described by Parseval's Theorem. Parseval's Theorem states that the total energy of a signal remains the same whether computed in the time domain or in the frequency domain.
- For a continuous-time signal \(x(t)\) and its Fourier Transform \(X(\omega)\), the energy \(E\) is given by:
- $$E = \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(\omega)|^2 d\omega$$
- For a discrete-time signal \(x[n]\) and its Discrete-Time Fourier Transform (DTFT) \(X(e^{j\omega})\), the energy \(E\) is given by:
- $$E = \sum_{n=-\infty}^{\infty} |x[n]|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(e^{j\omega})|^2 d\omega$$
These equations clearly show that the energy of the signal is conserved when it is converted into its Fourier representation. This means that the total "strength" or "content" of the signal, in terms of energy, does not change, only its representation shifts from time to frequency.
Power Conservation in Fourier Representation
Similar to energy, the average power of a signal is also conserved when it is transformed into its Fourier representation. For signals that have finite average power but infinite energy (e.g., periodic signals), we consider power conservation.
- For a continuous-time signal \(x(t)\) and its Fourier Transform \(X(\omega)\), the average power \(P\) is given by:
- $$P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt = \frac{1}{2\pi} \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |X(\omega)|^2 d\omega$$
- For a discrete-time signal \(x[n]\) and its Discrete-Time Fourier Transform (DTFT) \(X(e^{j\omega})\), the average power \(P\) is given by:
-
$$P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} S_{xx}(e^{j\omega}) d\omega$$
where \(S_{xx}(e^{j\omega})\) is the power spectral density.
The principle behind Parseval's theorem extends to power as well, ensuring that the average power of a signal remains invariant during the Fourier transformation. This is particularly important for analyzing the power distribution of signals in different frequency bands.
Summary of Conservation
Both energy and power are conserved properties when a time-domain signal is converted into its Fourier representation. This conservation is a fundamental aspect of Fourier analysis, allowing us to analyze signal characteristics equally well in either domain without loss of information regarding these key metrics.
- I. Energy: Conserved. Parseval's theorem directly states this.
- II. Power: Conserved. This also follows from the energy conservation principle, especially for power signals.
Therefore, both energy and power are conserved during the Fourier transformation process.
Important Questions from Fourier Series
If we use the Fourier transform ϕ(x, y) = \(\int {{{\rm{e}}^{{\rm{ikx}}}}} {ϕ _{\rm{k}}}\left( {\rm{y}} \right){\rm{dk}}\) to solve the partial differential equation \({\rm{ - }}\frac{{{\partial ^2}ϕ \left( {x,y} \right)}}{{\partial {y^2}}}\, - \,\frac{1}{{{y^2}}}\frac{{{\partial ^2}ϕ \left( {x,y} \right)}}{{\partial {x^2}}} + \frac{{{m^2}}}{{{y^2}}}ϕ \left( {x,y} \right) = 0\) in the half-plane {(x, y) : -∞ < x < ∞, 0 < y < ∞} the Fourier modes ϕ k(y) depend on y as y α and y β . The values of α and β are
The trigonometric Fourier series of a periodic time function can have
The Fourier series expansion of x3 in the interval −1 ≤ x < 1 with periodic continuation has
The Fourier series to represent x-x2 for –π ≤ x ≤ π is given by \(x - {x^2} = \frac{{{a_0}}}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}cosnx + \mathop \sum \limits_{n = 1}^\infty {b_n}sinnx\)
The value of a0 (round off to two decimal places), isThe discrete-time Fourier series representation of a signal x[n] with period N is written as \(\rm x[n] = \sum_{k = 0}^{N - 1} a_k e^{j(2kn\pi/N)}\). A discrete-time periodic signal with period N = 3, has the non-zero Fourier series coefficients: a- 3 = 2 and a4 = 1. The signal is
