Fourier transform of a real signal

In summary, a Fourier transform is a mathematical operation that breaks down a signal into its individual frequency components, allowing for analysis in the frequency domain. It differs from a Fourier series in that it is used for continuous and non-repeating signals, while a Fourier series is used for discrete and periodic signals. The Fourier transform of a real signal is complex because it represents both magnitude and phase, which are necessary for reconstructing the original signal. The inverse Fourier transform is used to convert a signal back to its original form in the time domain. The Fourier transform is a useful tool in signal processing, providing insights into frequency components, periodicity, and noise, and is used in applications such as filtering, compression, and modulation.
  • #1
FrogPad
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Taking a Fourier-transform of a real signal, gives me a spectrum that has symmetry.

If I take the FFT of a real signal, then throw away half of the spectrum, and then do an inverse transform I get a complex-signal.

I go from r(t) to rc(t) where rc(t) is a complex-signal.

Now this complex-signal supposedly contains all the information to reconstruct the original real-signal. My question is, how?
 
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  • #2
If anyone is interested, this question is answered in the following paper:
http://classes.engr.oregonstate.edu/eecs/winter2009/ece464/AnalyticSignal_Sept1999_SPTrans.pdf
 
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FAQ: Fourier transform of a real signal

What is a Fourier transform?

A Fourier transform is a mathematical operation that decomposes a signal into its individual frequency components. It allows us to analyze a signal in the frequency domain, which can provide valuable information about its characteristics.

What is the difference between a Fourier transform and a Fourier series?

A Fourier transform is used for signals that are continuous and of infinite duration, while a Fourier series is used for signals that are discrete and periodic. In other words, a Fourier transform is used for signals that are non-repeating, while a Fourier series is used for signals that repeat over time.

Why is the Fourier transform of a real signal complex?

The Fourier transform of a real signal is complex because it represents both the magnitude and phase of the signal's frequency components. The real part represents the magnitude, while the imaginary part represents the phase. This information is necessary to reconstruct the original signal.

What is the inverse Fourier transform?

The inverse Fourier transform is the mathematical operation that allows us to reconstruct a signal in the time domain from its frequency domain representation. It is the reverse of the Fourier transform and is used to convert a signal back to its original form.

How is the Fourier transform used in signal processing?

The Fourier transform is a powerful tool in signal processing because it allows us to analyze a signal in the frequency domain. This can provide insights into the characteristics of a signal, such as its frequency components, periodicity, and noise. It is also used in applications such as filtering, compression, and modulation of signals.

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