Recall the definition of the DTFT and the inverse DTFT:
$X(\omega) ~=~ \sum_{n=-\infty}^{\infty} x[n]\, e^{-j \omega n}, ~~~~~~ -\pi \leq \omega \lt \pi$
$x[n] ~=~ \int_{-\pi}^\pi X(\omega)\, e^{j\omega n} \, \frac{d\omega}{2\pi} , ~~~~~~ \infty\lt n\lt\infty$
We refer to a signal and its corresponding DTFT as a DTFT signal pair:$
x[n]~\stackrel{\mathrm{DTFT}}{\longleftrightarrow}~ X(\omega)
$
Dtft periodicity
Although the DTFT of a signal is typically defined only on a $2\pi$ interval, it is nevertheless straightforward to show that the DTFT is actually $2\pi$ periodic:
$X(\omega+2\pi k) ~=~ \sum_{n=-\infty}^{\infty} x[n]\, e^{-j (\omega+2\pi k) n}
~=~ \sum_{n=-\infty}^{\infty} x[n]\, e^{-j \omega n} \, e^{-j 2\pi k n}
~=~ X(\omega)~~\checkmark$
The $2\pi$ periodicity of the DTFT.
Dtft frequencies
Because $X(\omega)$ is essentially the inner product of a signal $x[n]$ with the signal $e^{j\omega n}$, we can say that $X(\omega)$ tells us how strongly the signal $e^{j\omega n}$ appears in $x[n]$. $X(\omega)$, then, is a measure of the "frequency content" of the signal $x[n]$. Consider the plot below of the DTFT of some signal $x[n]$:
This plot shows us that the signal $x[n]$ has a significant amount of low frequency content (frequencies around $\omega=0$), and less higher frequency content (frequencies around $\omega=\pm\pi, remember that the DTFT is $2\pi$ periodic).
As the DTFT is $2\pi$ periodic, all the information in a DTFT is contained in any $2\pi$ length section. However, there are only two intervals that are typically used. One is to refer to the DTFT from the interval of $0$ to $2\pi$. The low frequencies are at the extremities of the plot, with the highest frequency in the center:
The other commonly used interval chosen to express the DTFT is between $\pi$ and $\pi$. In that interval, the low frequencies are at the center, with the high frequencies at the edges:
The dtft and time shifts
If a signal is shifted in time, what effect might this have on its DTFT? Supposing $x[n]$ and $X(\omega)$ are a DTFT pair, we have that$
x[n-m]~\stackrel{\mathrm{DTFT}}{\longleftrightarrow}~ e^{-j\omega m} X(\omega)
$So shifting a signal in time corresponds to a modulation (multiplication by a complex sinusoid) in frequency. We can use the DTFT formula to prove this relationship, by way of a change of variables $r=n-m$:
$\begin{align*}\sum_{n=-\infty}^{\infty} x[n-m]\, e^{-j \omega n}&=
\sum_{r=-\infty}^{\infty} x[r]\, e^{-j \omega (r+m)}\\&=
\sum_{r=-\infty}^{\infty} x[r]\, e^{-j \omega r} \, e^{-j \omega m} \\[2mm]&=
e^{-j \omega m} \sum_{r=-\infty}^{\infty} x[r]\, e^{-j \omega r}\\
amp;=e^{-j \omega m} \, X(\omega) ~~\checkmark
\end{align*}$
The dtft and time modulation
We saw above how a shift in time corresponds to modulation in frequency. What do you suppose happens when a signal is modulated in time? If you guessed that it is shifted in frequency, you're right! If a signal $x[n]$ has a DTFT of $X(\omega)$, then we have this DTFT pair:$
e^{j\omega_0 n} \, x[n]~\stackrel{\mathrm{DTFT}}{\longleftrightarrow}~ X(\omega-\omega_0)
$Below is the proof:
$\sum_{n=-\infty}^{\infty} e^{j\omega_0 n} \, x[n]\, e^{-j \omega n}~=~
\sum_{n=-\infty}^{\infty} x[n]\, e^{-j (\omega-\omega_0) n}
~=~X(\omega-\omega_0)~~\checkmark
$
The dtft and convolution
Suppose that the impulse response of an LTI system is $h[n]$, the input to the system is $x[n]$, and the output is $y[n]$. Because the system is LTI, these three signals have a special relationship:$
y[n]~=~ x[n] * h[n]~=~ \sum_{m=-\infty}^{\infty} \: h[n-m] \, x[m]$
The output $y[n]$ is the convolution of $x[n]$ with $h[n]$. Just as with the other DTFT properties, it turns out there is also a relationship in the frequency domain. Consider the DTFT of each of those signals; call them $H(\omega)$, $X(\omega)$, and $Y(\omega)$. The convolution of the signals $x$ and $h$ in time corresponds to the multiplication of their DTFTs in frequency:
$y[n]=x[n]*h[n]\stackrel{\mathrm{DTFT}}{\longleftrightarrow} Y(\omega)=X(\omega)H(\omega)$
This relationship is very important. It gives insight, showing us how LTI systems modify the frequencies of input signals. It is also useful, because it gives us an alternative way of finding the output of a system. We could take the DTFTs of the input and impulse response, multiply them together, and then take the inverse DTFT of the result to find the output. There are some cases where this process might be easier than finding the convolution sum.
The dtft and linearity
Since the DTFT is an infinite sum, it should come as no surprise that it is a linear operator. Nevertheless, it is a helpful property to know. Suppose we have the following DTFT pairs:$x_1[n] ~\stackrel{\mathrm{DTFT}}{\longleftrightarrow}~ X_1(\omega) ~~~~~~~~x_2[n] ~\stackrel{\mathrm{DTFT}}{\longleftrightarrow}~ X_2(\omega)$Then by the linearity of the DTFT we have that, for any constants $\alpha_1$ and $\alpha_2$:
$\alpha_1 x_1[n] + \alpha_2 x[2]~\stackrel{\mathrm{DTFT}}{\longleftrightarrow}~ \alpha_1 X_1(\omega)+ \alpha_2 X_2(\omega)
$
Dtft symmetry
Looking at the plots of the DTFTs above, or perhaps of those you have computed on your own, you may notice that they are often symmetrical. It so happens that the DTFTs of real-valued signals always have a kind of symmetry, but there are other symmetry relationships as well. For example, signals that are purely imaginary and odd have DTFTs that are purely real and odd. These types of symmetry are a result of a property of the complex exponentials which build up the DTFTs. Any signal of the form $e^{j\omega n}$ is conjugate symmetric, meaning that its real part is even and its imaginary part is odd. Additionally, for conjugate symmetric signals, their magnitude is even and their phase is odd.
As a result of the symmetry of the signals which compose a DTFT, the DTFTs of certain signals also have symmetries, according to the table below:
***DTFT SYMMETRY TABLE HERE***
Questions & Answers
I'm interested in biological psychology and cognitive psychology
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
