We have seen that the discrete Fourier transform (DFT) is an example of an orthogonal basis for finite length discrete-time signals. As a basis, a linear combination of its basis elements $\{b_k\}$ can represent any signal $x$:
$x=\sum_{k=0}^{N-1}\alpha_k b_k=Ba$Furthermore, as an orthogonal basis, its basis elements are mutually orthogonal, meaning that the weights needed to build up a particular signal $x$ can be found with a weighted inner product of $x$ and the basis element in question. If all of the basis elements are unit norm, then the basis is orthonormal, and a simple inner product will find the weights:
$\alpha_k=\langle x, b_k\rangle~,~a=B^H x$The DFT is a particularly useful basis, for its basis vectors $e^{j\frac{2\pi}{N}kn}$ are the eigenfunctions of all LTI systems. As such, they are simply scaled when passing through a system. But as useful as the DFT is, there are other significant orthogonal bases to consider, as well.
The discrete cosine transform
Since the DFT uses complex harmonic sinusoids to analyze and synthesize signals, the DFT coefficients of real signals are almost always complex-valued (unless the signal is even, in which case the coefficients are purely real). This normally poses no problem, but there are some circumstances in which real coefficients are preferable. One example is when signal compression is desired. In such applications, a signal is transformed and only a few of the largest transform coefficients are kept. Having to store both the real and the imaginary parts of the coefficients would mean twice the storage.
It is for this reason that the discrete cosine transform (DCT) is used in signal compression applications (such as the JPEG image compression standard). The DCT is very similar to the DFT, as it also uses sinusoids as basis elements, except that its basis elements are purely real-valued. There are several varieties of the DCT, but the most popular is the DCT-II, which has the following basis elements:$d_k[n] ~=~ \cos\!\left[ \frac{\pi}{N} \left(n+\frac{1}{2}\right) k \right], \quad 0\leq n,k \leq N-1$
CAPTION. Unlike the DFT basis functions, the DCT functions are completely real-valued.
The real part of the DFT functions.The imaginary part of the DFT functions.The DCT functions are real-valued.CAPTION. Since the DCT basis functions are real valued, the DCT transform coefficients for a real signal are also real valued. As a result, there are cases in which signals can be represented reasonably well when reconstructed with just a handful of DCT coefficients, compared to the same number of DFT coefficients.
CAPTION.CAPTION.CAPTION.CAPTION.CAPTION.
Wavelet transforms
We have seen that with the DFT (and DCT as well), signals can be represented in either the time domain (e.g., as $x[n]$), or in the frequency domain through their DFT (e.g., $X[k]$). Both representations are equivalent, of course, for one can always be derived from the other. But sometimes one way of expressing a class of signals is desired, such as when as signal can be expressed with just a few values. For example, the signal $x[n]=\delta[n]$ can be expressed with just a single nonzero value in the time domain, but it does not have any nonzero values in the frequency domain. Thus we say the signal is localized in time. Likewise, the signal $x[n]=e^{j\frac{2\pi}{16}n}$ has only a single nonzero DFT coefficient, while all of its values in the time domain are nonzero. That is a kind of signal we say is localized in frequency. If a signal is localized in time or in frequency, we can efficiently store and process its signal values, by storing/operating on the signal in the best of those two domains.
But some signals are not localized very well in time or in frequency. If a signal has frequency components that show up and then are absent at different times, then neither the time or the frequency domain does a good job of efficiently representing such as signal. An example of such a signal would be a musical melody; at any given time it may be expressed as being a single (predominant) frequency, but these frequencies (i.e., the notes) obviously change quite a bit over time (the work of Philip Glass notwithstanding).
There are many transforms that are designed specifically for analysis and concise representation of these signals that have localized frequency components, two of which we will consider. The first is the Haar Wavelet, which is one of a class of wavelet transforms. The basis functions of the Haar wavelet transform, like other wavelets, are localized in both time and frequency, as illustrated in the following figure:
CAPTION.CAPTION.CAPTION.CAPTION. Unlike the DFT or DCT, wavelet basis functions are nonzero for most of their duration in time. But unlike delta functions, they have particular frequencies for their duration. So in a sense they have the benefits of both domains, combined.
The short time fourier transform
Another transform that considers signals which have time-localized frequency components is the short-time Fourier transform. The STFT is kind of like a DFT, except that it first splits a long signal into many smaller parts, and then takes a DFT of each of those parts. This "splitting" is done by multiplying the signal (of length $N$) by a moving window (of length $N_w$), progressively taking the DFT of the product:
$X[m,k]=\sum\limits_{n=0}^{N-1}x[n]w[n-m]e^{j\frac{2\pi}{N_w}k(n-m)$
The STFT of a signal is therefore usually represented as a two-dimensional image. Each column ($m$) of the image corresponds to a time window of a signal, while the rows ($k$) indicate how much of each frequency is present at a given time. It is therefore a bit like the sheet music score for a piano composition, in which the left-to-right axis of the music staff corresponds to time, while the markings up and down the staff correspond to the notes (i.e., frequencies) that are played at the given time.
CAPTION.CAPTION.CAPTION.
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.
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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
