# Using Linear Algebra for DSP

I am new to DSP in general, but can one use linear algebra by itself to characterize a signal?

My first idea was to transform the signal into a matrix and then use the determinate to characteristic the signal. The big problem being that matrices obey row echelon form which means the rows can be switched and produced the same determinate.

• hm, I work in an institute that has done a lot of work on signal processing for radio signal processing. I'd go as far as saying that LA is our main weapon. I hear myself saying in exercises that "we're the institute for multiplication and addition, so get used to us solving things by base transforms making things orthogonal". So, yeah, one can use linear algebra to characterize a signal. As Fat32 notices, the DFT is a linear transform, but we use LA to describe signals as elements from vector spaces all over the place. Compressed sensing, CDMA, a lot of parametric spectrum estimators,… all LA Commented Jul 19, 2019 at 8:54

Yes indeed modern signal processing uses matrix methods of linear algebra (or Linear System Theory more correctly), in addition to the classical calculus and harmonic analysis.

Linear algebra (matrix) based DSP considers (1D) signals as vectors and transforms as matrices that map inputs to outputs. As CedronDawg also mentioned, in such a representation the DFT (discrete Fourier transform) is merely a matrix mapping:

$$Y = \Phi X$$ where $$X = [x_1,x_2,...,x_N]^T$$ is an $$N \times 1$$ input vector, $$Y = [y_1,y_2,...,y_N]^T$$ is the output vector, and $$\Phi$$ is the $$N \times N$$ DFT transform matrix.

For a brief treatment of matrix methods in DSP, have a look at the books:

• Multiresolution Signal Decomposition_AKANSU

• Statistical Digital Signal Processing_HAYES

• I find it interesting you put the modifier "classical" in front of "Calculus". Not sure what you mean by that. The Calculus that is used to derive the DFT from the FT requires Real Analysis. Indeed, in my RA classes we were told that it was the problem of whether a sequence continuous series (e.g. the partial sums of Fourier Series) could converge to a discontinuous function was the impetus for the creation of the field. I'll admit, not my favorite math. I was also a Calculus TA in college, and by no means am I trying to discourage its study. Commented Jul 19, 2019 at 1:03
• @CedronDawg by the advent of digital computer, computation emphasis has been the matrix algebra and numerical methods. I used classical to refer to anything not using computers :-) Commented Jul 19, 2019 at 1:12
• Okay. When I hear "classical Calculus" I think "Newtonian Calculus". I would point out that both Calculus and Linear Algebra were well mature before the advent of even primitive computers. Math theory always seems to be ahead of any implementations. Go figure. Commented Jul 19, 2019 at 2:30

Actually, considering a signal to be a vector, or a N x 1 matrix, makes the DFT the same as a multiplication of a matrix.

$$X[k] = \sum_{n=0}^{N-1} x[n] e^{i\frac{2\pi}{N}nk }$$

$$X = F x$$

In many ways, this is the best paradigm to characterize the behavior of the DFT. The proper choice of normalization ($$\frac{1}{\sqrt{N}}$$) makes it a unitary matrix.

Linear Algebra is also key to several of my tone parameter calculation formulas.

Linear Algebra is not limited to these applications in DSP by any means. You will often see "Linear Algebra" mentioned whenever anybody asks "What math should I learn to help me with DSP".

Way more important than Calculus. Calculus is not even needed to understand the DFT.

Representing a "signal" as a matrix is not that meaningful to me.

• Calculus is useful in digital signal processing as we often use discrete integration or derivative. For example, when designing a software PLL, understanding that angular frequency is the derivative of the phase is essential. It is not required for the DFT I agree, but if you're the kind of person to dig into the entrails of the DFT/FFT, you're likely to have a good math background.
– Ben
Commented Jul 19, 2019 at 15:44
• @Ben A minor correction: There is no such thing as "discrete integration or derivative", you can approximate integrals and derivatives discretely. This is the very foundation of Calculus, yes? You are correct, I have a degree in Math and a lifelong interest. I've been neck deep in entrails for the past dozen years trying to convince people that integration is based on summation (and exact solutions are possible), not the reverse and that applies to the Fourier Transform too. To me, the Calculus in DSP is only applicable to ADC and DAC. Once you've gone "Digital" it's all discrete. Commented Jul 19, 2019 at 16:00
• Perhaps I should have said that concepts seen in calculus are similar to concepts seen in digital signal processing and I feel like calculs helps me understanding digital signal processing concepts better.
– Ben
Commented Jul 20, 2019 at 0:10
• I knew what you meant, and I meant no offense. And I agree. Calculus is helpful in the understanding of all sorts of concepts. But there are newbies around and misleading language isn't helpful to anyone. My last sentence was a little extreme too. There are definitely advantages of having Calculus in your toolbox when you are considering a discrete signal as representing a continuous one. Sorry, I didn't mean to sound so harsh. Commented Jul 20, 2019 at 1:03
• None taken. Don't worry.
– Ben
Commented Jul 20, 2019 at 2:01