# use of complex number in digital signal processing [closed]

how complex number used in digital signal processing. detailed information about on how this method will be useful in radar technology.

• I feel like the last question, i.e. the general question about radar is something you could look up easily on a website like Wikipedia. Have you tried searching for information on what a radar is and what it does? Perhaps then it might help you understand why complex numbers are important Aug 26, 2019 at 12:34
• ok, Thank you for your suggestion Aug 27, 2019 at 4:06

Complex numbers have a lot of applications in DSP in general, but since you specifically asked about radar systems, I can tell you why we use complex numbers in radar signal processing (all of my professional work is in this field).

You'll often see collections of complex numbers in radar signal data referred to as "in-phase/quadrature" data, or IQ data. National instruments has a nice little tutortial that explains it well with pictures, you can see that here.

It basically boils down to this: for radar, we care a great amount about retrieving the phase of a signal. Without knowing the phase, an ambiguity exists in frequency: we cannot determine (without phase) if a frequency is negative or positive. This leads to ineffective bandwidth usage at what is called baseband. By using IQ, i.e. complex data, we can break this ambiguity and have a unique frequency spectrum with no mirroring or redundancies. This is nice from a processing standpoint, as it generally affords us the ability to use half of sampling rate normally required to processing a signal that wasn't at baseband.

Another reason for using complex data in radar is again rooted in the concept of recovering phase. Pulse Doppler processing requires observations of phase changes so that we can determine a target's velocity. In order to track the target's phase change, we of course have to be able to track its phase to begin with. As you'll see in that national instruments tutorial, by using complex data we are able to achieve the maximum likelihood estimate of phase, and thus we can use pulse Doppler processing to generate a range-Doppler map, allowing for detection of a target in both range AND velocity, which can be very useful for specific applications.

In summary, using IQ data gives us an additional degree of freedom such that we can recover phase, which for most radar signal processing is incredibly useful, as it makes signals more easily separable and enables certain types of processing.

• Thank you for your help Aug 27, 2019 at 4:54

Any complex array could be represented as a real array as follows: $$x_R = \begin{bmatrix} \Re(x) \\ \Im(x) \end{bmatrix}$$ and matrices are represented as $$A_R = \begin{bmatrix} \Re(A) & \Im(A) \\ -\Im(A) & \Re(A) \end{bmatrix}$$ where $$\Re(x),\Im(x)$$ are vectors containing the real and imaginary part of complex vector $$x$$, respectively. Note that addition/subtraction is preserved by simply addition $$x_R + y_R = z_R$$ You can also verify that matrix multiplications in the real domain also preserves the real/complex information in their respective blocks, namely $$A_RB_R = C_R$$ where $$C_R = \begin{bmatrix} \Re(C) & \Im(C) \\ -\Im(C) & \Re(C) \end{bmatrix}$$

• please can you explain in more detail Aug 26, 2019 at 12:16