# Polyphase components Spectrum formula in Discrete time

I am trying to understand the meaning behind some of the components of the formula for polyphase components in discrete time. Note that $$\Omega$$ indicates discrete in this notation and the formula is given by:

Polyphase components is dividing the spectrum into blocks of size $$L$$ and each block divided into $$k$$ parts. If we remove the exponentials $$e$$ we have the formula for downsampling which makes sense since we are getting rid of the other polyphase components. The first exponential $$e^{jk \Omega/L}$$ seems to be the shift to move the index $$k$$ of the block of size $$L$$. However, I do not know what is the purpose of the second exponential $$e^{-2\pi jkp/L}$$

Thank you

$$P_k\left( e^{j\Omega} \right) = \frac{1}{L} e^{jk\Omega/L} \sum_{p=0}^{L-1} e^{-2 \pi j k p / L} H\left( e^{j(\Omega-2\pi p)/L} \right)$$

Do a little rearranging.

$$P_k\left( e^{j\Omega} \right) = e^{j\Omega\frac{k}{L}} \frac{1}{L} \left[ \sum_{p=0}^{L-1} H\left( e^{j\Omega\frac{1}{L} } e^{-j2\pi \frac{p}{L}} \right) \left( e^{j2\pi/L} \right)^{-kp} \right]$$

It can be made simpler to understand by a few variable substitutions.

$$y = e^{j\Omega}$$

$$y$$ is a complex value on the complex unit circle at $$\Omega$$ radians around the circumference.

$$h[p] = H\left( y^{\frac{1}{L}} e^{-j2\pi \frac{p}{L}} \right)$$

$$h[]$$ is a sequence of values sampled from the $$H()$$ function. $$y^{\frac{1}{L}}$$ is a complex value on the unit circle at $$1/L$$th the angle of $$y$$. The $$e^{-j2\pi \frac{p}{L}}$$ is a clockwise walk around the unit circle in $$L$$ even steps. Conceptually:

$$e^{-j2\pi \frac{p}{L}} = \left( e^{j2\pi } \right)^{-\frac{p}{L}} = 1^{-\frac{p}{L}}$$

Therefore $$h[]$$ has a period of $$L$$. If $$p$$ were to go below zero or above $$L-1$$, it would be a repeat pattern.

With these substitutions, your equation becomes this:

$$P_k\left( y \right) = y^{\frac{k}{L}} \frac{1}{L} \left[ \sum_{p=0}^{L-1} h[p] \left( e^{j2\pi/L} \right)^{-pk} \right]$$

The part in brackets is the definition of the DFT.

$$X[k] = \sum_{n=0}^{N-1} x[n] \left(e^{i2\pi/N}\right)^{-nk}$$

It's arranged the same way as in my answer here Fractional powers of complex numbers (DSPrelated computation)

The $$\frac{1}{L}$$ is a normalizing factor making the magnitudes of the DFT calculation independent of $$L$$.

$$y^{\frac{k}{L}}$$ is a complex value $$k/L$$ of the way to $$y$$ along the unit circle which means the results of the DFT bin are rotated that far.

So, a long answer to your question. The purpose of the second exponential $$e^{-2\pi j k p/L}$$ is to execute the DFT transform. It is part of the definition and the reason it works the way it does. Lots of stuff available on that if you are unfamiliar.