Transforming Real Signals to Complex Representation

In summary: I guess any real signal can then be viewed as a complex function, meaning that it is equivalent to its Fourier transform, which is a complex function...Yes, you can always view a real signal as a special case of a complex signal, where the imaginary part is zero. Sometimes it is useful to think of a real signal as a complex signal with a zero "imaginary" part. The "complex envelope" notation is often used in textbooks on communications, radar, etc., to make the math easier to follow. You can always recover the real signal from the complex envelope.In summary, a generic, not necessarily harmonic, signal of time can be represented as a complex signal with a real and imaginary part.
  • #1
fisico30
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Can a generic, not necessarily harmonic, signal of time be represented as a complex signal with a real and imaginary part?
Usually the complex rappresentation is used for time harmonic signals and linear systems.
The "real" time signal is transformed into a complex signal. At the end of the mathematical operations, it is possible to look at the real or imaginary part of the complex signal and get the correct results for the initial signal.
But what if the starting real signal is not harmonic?

thanks!
 
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  • #2
We can almost always represent a signal in the frequency domain using Fourier transformations or series. In this case the signal would be a finite or infinite set of harmonic signals.
 
  • #3
Yes, this is done all the time in communications, radar, etc., for non-harmonic signals. Essentially, if you have a real signal, then the negative frequency Fourier components are the complex conjugate of the positive frequency components. Thus you only need half of the spectrum - setting the negative half to zero leaves you with a complex signal. In applications we always have band-limited signals and what is usually done is as follows:

1. take the time-domain signal and multiply it by a phase ramp, this shifts the frequencies. Often we shift the frequency so that the middle of our band is centered on DC. This shifts the other half of the spectrum to higher frequency.

2. Low-pass filter to get rid of the half of the spectrum that is at the higher frequency.

Now you have a complex signal, which has less total bandwidth than the original signal. Since you know what operations you did on the signal, you can always reconstruct the original real signal.

In the olden days this was done analog - take a signal, split it, mix one path with sin(wt) and the other with cos(wt), then use two A\D converters, one for each channel. One channel is "real" the other is "imaginary". What you save is that the total bandwidth is smaller so your required sample rate is lower (often giving you better dynamic range, less power consumption, etc.). These days we often do all this digitally. After the low-pass filter you can usually do some decimation to reduce the amount of computation required further up the chain.

If you want more details, google "IQ sampling" and you will likely find more info. Communications books usually have nice discussions of this stuff, as do radar books.
 
  • #4
Hi JasonRF,

thanks for the good response. Just one last clarification.
A real signal has a Fourier transform which is a complex function of the real variable frequency. This function exists for the negative and positive frequency.
By realizing that the Fourier transform is not only a complex function but also Hermitian, we can get rid of the negative frequencies (since their information is already contained in the positive frequencies). What we get is the complex analytic signal. What is the advantage of this operation? What do we gain? Are we saving bandwidth?
There is something about the Hilbert transform here too...
I guess any real signal can then be viewed as a complex function, meaning that it is equivalent to its Fourier transform, which is a complex function...
thanks
fisico30
 
  • #5
fisico30 said:
Hi JasonRF,

thanks for the good response. Just one last clarification.
A real signal has a Fourier transform which is a complex function of the real variable frequency. This function exists for the negative and positive frequency.
By realizing that the Fourier transform is not only a complex function but also Hermitian, we can get rid of the negative frequencies (since their information is already contained in the positive frequencies). What we get is the complex analytic signal.

Exactly!

fisico30 said:
What is the advantage of this operation? What do we gain? Are we saving bandwidth?
Yes, you effectively save bandwidth. Once you only have 1/2 the spectrum, you can shift it in frequency (by multiplying by a phase ramp) to be centered at DC. You can decimate, and perform filtering, etc., on the complex signal. It also can make it "easy" to separate signals out by phase: applying phase shifts to complex signals is easy! So this is a practical thing to do, and is done all the time in communications equipment, radars, etc. It also makes the theoretical analysis of such systems easier, since we can just deal with the complex envelope, below ...

fisico30 said:
There is something about the Hilbert transform here too...
Yes. If you have a real signal [tex]x(t)[/tex], then taking the Hilbert transform of the signal is equivalent to multiplying the Fourier transform of [tex]x(t)[/tex] by [tex]-i sgn(f)[/tex], where [tex]sgn(f)[/tex] is the "sign" function that is 1 for positive f and -1 for negative f. If the signal [tex]x(t)[/tex] is bandpass with center frequency [tex]f_c[/tex], then the complex envelope is usually written

[tex]\tilde{x}(t)=e^{-i2\pi f_c t}\left[ x(t)+i\hat{x}(t)\right][/tex],

where [tex]\hat{x}(t)[/tex] is the Hilbert transform of [tex]x(t)[/tex].
The Fourier transform of the portion in brackets is zero for negative frequencies, and is twice the Fourier transform of [tex]x(t)[/tex] for positive frequencies. The phase ramp out in front simply shifts the remaining part of the spectrum to be centered at DC. You recover the original signal simply with

[tex] x(t) = Re \left[\tilde{x}(t) e^{i 2 \pi f_c t} \right] [/tex].

Just like for harmonic signals, this simplifies analysis. If we want apply a filter with an impulse response [tex]h(t)[/tex], make the complex envelope version (with an extra 2 to make the end result nicer) such that


[tex] h(t) = Re \left[2 \tilde{h}(t) e^{i 2 \pi f_c t} \right] [/tex].

Then, the convolution

[tex]y(t) = h(t) \ast x(t)[/tex]

can be written

[tex] y(t) = Re \left[ \tilde{y}(t) e^{i 2 \pi f_c t} \right] [/tex]

where

[tex]\tilde{y}(t) = \tilde{h}(t) \ast \tilde{x}(t)[/tex].

For analytical work this is much easier to deal with!
 

FAQ: Transforming Real Signals to Complex Representation

1. What is the purpose of transforming real signals to complex representation?

The purpose of transforming real signals to complex representation is to simplify the analysis and processing of signals. Complex representation allows for the use of mathematical tools such as the Fourier transform, which can provide insights into the frequency components and characteristics of a signal.

2. How is a real signal transformed into complex representation?

A real signal can be transformed into complex representation using the technique of analytic signal construction. This involves taking the original signal and adding a 90-degree phase-shifted version of itself, resulting in a complex signal with both real and imaginary components.

3. What are the benefits of using complex representation in signal processing?

Complex representation allows for the use of powerful mathematical tools such as the Fourier transform, which can provide a more detailed analysis of a signal's frequency components and characteristics. It also simplifies many signal processing operations, such as filtering and modulation.

4. Are there any limitations to transforming real signals to complex representation?

One limitation is that the transformation process may introduce errors or distortions in the signal, which can affect the accuracy of the analysis. Additionally, complex representation may not be suitable for all types of signals, such as non-stationary or non-linear signals.

5. Can complex representation be applied to any type of signal?

Complex representation can be applied to a wide range of signals, including audio, video, and communication signals. However, it may not be suitable for all types of signals, such as non-stationary or non-linear signals. It is important to consider the characteristics of the signal and the intended analysis or processing when deciding whether to use complex representation.

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