# Understanding graphs of DTFT with time shift of$~y\left[n\right]=x\left[n-2\right]~$

$$x\left[n\right]:=\text{discrete time signal}\tag{1}$$

The following plot is DTFT of$$~x\left[n\right]~$$

What I know so far are as below.

$$x\left[n\right]=\frac{1}{2\pi}\int_{0}^{2\pi}X\left(\exp\left(j\omega\right)\right)\exp\left(j\omega n\right)\,d\omega_{}\tag{2}$$

$$X\left(\exp\left(j\omega\right)\right)=\sum_{n=-\infty}^{\infty}x\left[n\right]\exp\left(-j\omega n\right)\tag{3}$$

I know the following property of DTFT.

$$\underbrace{x\left[n-k\right]}_\text{time domain}~~\leftrightarrow~~\underbrace{X\left(\exp\left(j\omega\right)\right)\exp\left(-j\omega k\right)}_\text{frequency domain}\tag{4}$$

The below plot is of DTFT of signal$$~y\left[n\right]=x\left[n-2\right]~$$

I can't get why right side takes$$~\theta_{}\left(\omega_{}\right)=-2\omega_{}~$$

Moreover, I even can't get why left side diagram can be held, since the following is held.

$$x\left[n-2\right]~~\leftrightarrow~~X\left(\exp\left(j\omega\right)\right)\exp\left(-j\omega 2\right)\tag{5}$$

What should I study first?

First, you seem to be lacking some basics and should read a standard text book on the topic, e.g. the ever mentioned Oppenheim et al. "Discrete-time signal processing".

• Your nomenclature of the transform is kind of confusing. Just call it $$X(\omega)$$.
• You correctly state in $$(3)$$, that a shift in time domain is equivalent to a modulation in frequency domain. A modulation means, mathematically, a multiplication by a complex pointer, which purely affects the phase of the signal. An alteration of a signal's phase cannot be seen in its magnitude plot but only in its phase plot which is the diagram on the right side.
• The magnitude plot shows triangles. Referring to the table of basic transform pairs, the signal is of the $$\text{sinc}^2$$ category, of which $$\theta$$ is no function of $$\omega$$. So the pase of $$x[n-2]$$ can be read directly from the exponent.