Differences between Frequency Domain and Time Domain
The time-domain representation gives the amplitudes of the signal at the instants of time during which it was sampled. However, in many cases you need to know the frequency content of a signal rather than the amplitudes of the individual samples.
Fourier's theorem states that any waveform in the time domain can be represented by the weighted sum of sines and cosines. The same waveform then can be represented in the frequency domain as a pair of amplitude and phase values at each component frequency.
You can generate any waveform by adding sine waves, each with a particular amplitude and phase. The following figure shows the original waveform, labeled sum, and its component frequencies. The fundamental frequency is shown at the frequency f0, the second harmonic at frequency 2f0, and the third harmonic at frequency 3f0.
In the frequency domain, you can separate conceptually the sine waves that add to form the complex time-domain signal. The previous figure shows single frequency components, which spread out in the time domain, as distinct impulses in the frequency domain. The amplitude of each frequency line is the amplitude of the time waveform for that frequency component. The representation of a signal in terms of its individual frequency components is the frequency-domain representation of the signal. The frequency-domain representation might provide more insight about the signal and the system from which it was generated.
The samples of a signal obtained from a DAQ device constitute the time-domain representation of the signal. Some measurements, such as harmonic distortion, are difficult to quantify by inspecting the time waveform on an oscilloscope. When the same signal is displayed in the frequency domain by an FFT Analyzer, also known as a Dynamic Signal Analyzer, you easily can measure the harmonic frequencies and amplitudes.
Parseval's Theorem
Parseval's Theorem states that the total energy computed in the time domain must equal the total energy computed in the frequency domain. It is a statement of conservation of energy. The following equation defines the continuous form of Parseval's theorem.
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(A) |
The following equation defines the discrete form of Parseval's theorem.
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(B) |
where 
is a discrete FFT pair and n is the number of elements in the sequence.
The following figure shows the block diagram of a VI that demonstrates Parseval's theorem.
The VI in the previous figure produces a real input sequence. The upper branch on the block diagram computes the energy of the time-domain signal using the left side of Equation B. The lower branch on the block diagram converts the time-domain signal to the frequency domain and computes the energy of the frequency-domain signal using the right side of Equation B.
The following figure shows the results returned by the VI in the previous figure.
In the previous figure, the total computed energy in the time domain equals the total computed energy in the frequency domain.
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