# DFT symmetry vs DFT duality in Richard Lyons' "Understanding DSP"

I am reading Richard lyons, understanding dsp, chap 3.

Article 3.2 is about property of dft symmetry but any where in this chapter, i am unable to find discussion about dft duality property

I want to know here ,author means both dft symmetry and dft duality to be a same thing?

• I don't have the book at hand (Corona has me far away from our library), and I'm not aware of a clear definition of "DFT duality". Could you add Lyons' defintion of "DFT symmetry" and your definition of "DFT duality" to your question? Without knowing your specific definition of duality, it will be hard to make any statement, even when knowing Rick's book by heart (he's a user on this site). Jun 5, 2020 at 23:21
• ("DFT" implies "in the context of DSP", so I changed the title, hopefully it helps.) Jun 5, 2020 at 23:21
• @DSPRookie has added the Lyons definition of DFT symmetry. Could you confirm that their definition matches what you had in mind? That would be awesome. Jun 6, 2020 at 9:28

## 1 Answer

1. Duality between discrete frequency and discrete time domain.

DFT Duality is generally referred to the duality of DFT-IDFT pairs. This in turn comes from the similarity between analysis and synthesis expressions of DFT and IDFT. $$X[k] = \sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi}{N}nk}$$ $$x[n] = \frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{j\frac{2\pi}{N}nk}$$ (Although I will encourage you to understand DFT as projection of Finite-length sequence x[n] onto Discrete Fourier Orthogonal Basis Vectors, and IDFT as representation of x[n] as linear combination of those orthogonal fourier basis vectors, where coefficients of linear combination is are the DFT coefficients $$X[k]$$. You might want to read this answer : DFT as Projection on Orthogonal Fourier Basis Vectors)

By Duality between the DFT-IDFT pairs I mean, for example, DFT of discrete $$\delta[n]$$ is a constant $$1$$ in freq-domain and DFT of constant $$1$$ will be discrete $$\delta[k]$$ in freq-domain. And other example would be DFT of discrete $$rect$$ sequence i.e. sequence of $$M$$ $$1$$'s in an $$N>M$$ length sequence. It has its DFT as follows: $$X[k] = e^{-j\frac{\pi}{N}(M-1)k}\frac{\sin[\frac{\pi}{N}Mk]}{\sin[\frac{\pi}{N}k]}, \forall \ k \in \ \{0,1,2,...,(N-1)\}$$ And, if we took DFT of $$x[n] = e^{-j\frac{\pi}{N}(L-1)n}\frac{\sin[\frac{\pi}{N}Ln]}{\sin[\frac{\pi}{N}n]}$$, we will get a sequence of $$L$$ consecutive $$1$$'s in an $$N$$ length DFT. We can just run the steps backwards to compute the DFT of this $$x[n]$$.

For continuous time fourier transform, duality would mean the following : if $$f(t)$$ has fourier transform $$F(\Omega)$$ then a time domain function $$F(t)$$ will have its fourier transform as $$2\pi f(-\Omega)$$.

2. Symmetry in Discrete Fourier Transform Coefficients.

DFT Symmetry in the book is mentioned in context of symmetry properties in DFT coefficients when DFT of a real valued time domain sequence $$x[n]$$ is computed. That is DFT coefficients of all real valued $$x[n]$$ are conjugate symmetric modulo $$N$$. $$X[k] = X^*[(N-k) \ \mod \ N]$$

• well answered, concise Jun 6, 2020 at 6:31
• Ah, this is really interesting! It's quite unlike any "duality" term I was expecting to be applied: in math, duality usually means that there's a structure-preserving mapping from one thing to the other: For example, the DFT as operation has a group homomorphism $(V, *) \mapsto (V^*, \cdot)$ with $*$ being circular convolution in the original vector space, and $\cdot$ the point-wise multiplication in the image space. But there's other duality terms, too, like one defined for bilinear operations, and generally, one could argue that every isomorphism (like the DFT) implies duality. Jun 6, 2020 at 9:03
• Hence, I'm really surprised to see a "oh this formulas just look similar" definition, but I bet it makes sense when one takes a closer look. Can you tell me in which context you've learned this? More of the communications context or more of the finite spacey physical context, or maybe more on the radar context? Jun 6, 2020 at 9:05
• @MarcusMüller I remember this idea of duality from very long back when I first learnt properties of Fourier Transform (May be not DFT, but continuous time fourier transform). And since the OP has been reading basic textbook on DSP, so I thought of giving the answer in the same context. Duality in Fourier Transform in many initial level is taught like: if fourier transform of $f(t)$ is $F(\omega)$, then fourier transform of $F(t)$ will be $2\pi f(-\omega)$. Like, if fourier transform of rect is sinc, then fourier transform of sinc is rect, etc. Jun 6, 2020 at 9:24
• @DSPRookie thanks! So that's pretty nice a concept :) Thanks for taking the time to reply. Jun 6, 2020 at 9:27