# Can we generate signals in reality that have non symmetric magnitude spectrum in the Fourier domain?

I am reading communication systems, and have a doubt: Is it possible to generate signals in reality that have a non-symmetric magnitude spectrum in the Fourier domain? For example, if I have a signal $$u(f)$$ or something like this as Fourier domain representation

I know that their Inverse Fourier transform is going to have imaginary element $$j$$ in them, but does this $$j$$ just symbolize 90-degree phaseshift, can I generate signals in reality that have asymmetric magnitudes of their Fourier transforms.

I have read that $$j/-j$$ just represent a 90-degree lag or lead, but do such signals with asymmetric magnitude spectrum in the Fourier domain exist in reality? This will also clear my doubt that whether there are such signals in reality that have just positive/negative frequency components in their Fourier domain representation.

As the answers suggest having $$j$$ just means phase shift, but if some practically permissible signal has representation $$jm(t)$$, where $$m(t)$$ is real, with no imaginary component, then with respect to what I have to define the phase.

Or such a condition will never arise where the inverse Fourier transform of a signal is purely an imaginary quantity.

If you have a 2 dimensional signal with orthogonal components, then sure. Just call one of the dimensions the imaginary component from the complex result of an inverse FFT or DFT. This can be a close approximation to many pairs of physical measurements, such as voltage and current in certain topologies of AC electrical circuits, or single-sideband baseband IQ signal pairs (two quadrature voltages representing one SSB signal after balanced mixer modulation to RF).

For any 1 dimensional real signal, then no; as the FFT spectrum of any 1 dimensional real signal will always be perfectly conjugate symmetric with cancelling mirror symmetric imaginary components and symmetric (thus equal) real components. You can change the phase(s) any way you want, and the FFT will still be conjugate symmetric. If you change a signals phase, only the angle between the conjugate symmetric vectors will change, but the imaginary parts will still cancel out to zero.

• Can you please check my edits to the question, those with bold fonts, I will be grateful. May 25 '21 at 19:26

As @Bob said, it depends of your definition of "in reality". I am going to ignore images on this post and focus on signals as a function of time.

• One-dimensional signals always have a symmetrical magnitude spectrum. Examples are the voltage or current on a wire, or a sequence of real numbers stored in digital memory.

• Two-dimensional signals may have a non-symmetrical magnitude spectrum. Examples are complex signals, whether defined mathematically or stored in a digital memory; and the voltage or current on two wires, one of which is labeled $$\sqrt{-1}$$ and with the agreement of doing complex arithmetic (see $9\rm V$ Battery with $45^\circ$ phase).

In my opinon, all these signals exist in reality, and then the answer is that yes, some signals in reality may indeed have a non-symmetrical magnitude spectrum.

• Basically, this means that all analytic signals are physical signals available in nature? May 25 '21 at 18:55
• Yes, as long as you accept human-generated signals. I don't know of any signal with non-symmetric spectrum that is not generated by humans.
– MBaz
May 25 '21 at 18:59
• I have just one more query if a signal is not two-dimensional but just have a $j$ term in it, for example, $j$m(t), where m(t) is real, then how will I define its phase, means with which respect, I have to define its phase. How will this signal be generated? May 25 '21 at 19:12
• @UserHuffmann you just apply the usual formula: the angle is the arctan of the real part divided by the imaginary part. A purely imaginary signal has phase $\pi$, just like a purely real signal has phase $0$.
– MBaz
May 25 '21 at 22:05

If by "signal in reality" you mean a real signal, than the answer is no.

You can just look at the definition of the Fourier transform

$$X(\xi) = \int x(t) e^{-j \xi t} dt$$

and see that $$X(-\xi)$$ is the complex conjugate of $$X(\xi)$$, therefore, the frequency response will be symmetric.

If by "signal in reality" you mean a physical signal, then yes it is possible if you have a signal that is represented as a phasor. Common practice in communications where you perform an IQ detection.