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Conditional entropy $H(X|Y)$ tells us how much the average uncertainty about a channel input $X$ is after observing channel output $Y$, and mutual information $I(X,Y)$ measures how much information about channel input $X$ can be obtained observing channel output $Y$. Doesn't this mean that $H(X|Y)$ and $I(X,Y)$ complem...
I am wondering about whether or not there is a theoretical limit as to the speed at which we can compute a DFT. We all know that the FFT executes in $O (N \log N)$ time. However, is this a lower bound of some sort? Might there be yet faster algorithms that have yet to be discovered which we do not currently know about...
I am using a miniature car and I want to estimate the position. We can not use GPS modules and most of the tracking systems that I saw, are using IMU sensor with the GPS module. In our car we are able to find our exact correct location with image processing but for some parts that don;t have enough markings we can not ...
I understand and have implemented a 2D discrete wavelet transform that uses a Haar wavelet to process images. Now I would like to implement a 2D Cohen-Daubechies-Feauveau wavelet transform, but I am uncertain about how to do the inverse transform. The Haar transform uses a 2x2 kernel, so I have 4 coefficients for each ...
First of all I would like to say that : I'm new to DSP and in a "learning curve" hence the "basicness" of my questions. I've read as many posts/articles as possible on the internet but still need "guidance". I'm french hence the bad formulation sometimes. My mission/data context : Performing data analysis on real dat...
I am fairly new to DSP have been reading up on digital filters and have a few questions. I have a signal with very high frequencies whose DC component i want to estimate. The magnitudes of these frequencies can be very high too. I can average out the signal, and this works, but the high frequencies mean high sample rat...
Using MATLAB signal processing toolbox, I have created signals like dirichlet, chirp and etc and and obtained their fourier and cosine fourier transform. Now I want to find the frequency where for example A% frequencies are less than that and use it as a thresholding frequency to compact the signal and eliminate unnece...
I'm trying to understand PSD, and, it's functions. So far I have carried out the following: Segmented a signal with NFFT = 256 and hop_size = 128 Windowed each of these segments (Hamming) DFT of each Window Taken the first half of the resulting matrix (n/2+1) From what I have read so far, the algorithm can be best de...
I'm trying to figure out how FFT's phase works. Here's my code: import numpy as np t = np.linspace(0,120,1200) acc = lambda t: 10*np.sin(2*np.pi*t) # signal is a perfect 10 amplitude 1 frequency sin wave signal = acc(t) FFT = np.fft.rfft(signal) freqs = np.fft.rfftfreq(len(signal),1/10.) real = signal[110] total = 0...
I have a doubt, it may be silly. Please explain me. In K-L Transform, For forward transform we have the below equation For inverse transform we have the below equation In reverse transform why the 'A' transpose is not common to both y and m(mean of x) as in Forward transform equation?
I am running an acoustic simulation. I put in a frequency and get out a complex number; the frequency response for that frequency. I wish to generate a 1024-sample impulse response (supposing my output sample-rate is 44.1kHz) How do I go about it? I think I have the first step: take the Nyquist frequency 22.05kHz, and...
I am a quite new to the area of digital signal processing. I am currently doing a project on sine wave generation. I know that according to Gibbs phenomenon, many sine waves add up to form a square wave. So, square wave would contain many frequencies. If we filter that square wave stage by stage using low pass and band...
I'm learning about the Wiener filter, and I'm working on my own implementation. I'm starting out with the non-causal filter, and I need to calculate some covariance matrices. If the signal is assumed to be a linear combination of a target signal and noise, then the signal can be modeled as: $$x[n] = y[n] + w[n]$$ where...
I am I am working with 'wav' files in audio signal processing. I am able to find out STFT by using the function " spectrogram(file name, window length). Now I want to apply Inverse Short time fourier transform on the above signal. I tried "invspectrogram". but its showing error as Undefined function for input argument ...
I know we can use a -step -impulse -sinus -multisine -random phase multisine To identify a system. But what are the advantages/disadvantags of each type of input?
I want to know what is the difference between additive noise, multiplicative noise.. In what domain these noises are handled to remove? I am interested particularly in image dataset Does different domains are used in case of satellite imaging where noise due to reflection of frequencies is statistically independent?
This question comes from a realistic problem that how to obtain a time domain wind pressure from known wind spectrum. Or similarly, gets rail surface curve from spatial rail spectrum. Obviously, this question does not have unique solution, since phase information is not given. Usually we assume that phase of the time/...
I just read the article linked to below: http://www.eetimes.com/document.asp?doc_id=1275520 Now, I'm planning on doing a Masters in Signal Processing. I'd like to know what the career prospects are for someone graduating in this specialization. Note that I live in the high-tech capital of India, so I should be in abou...
I am reading a paper using log-gabor filters for feature detection. I was thinking about the difference between Gabor filters and log-gabor filters. Can anyone tell me the difference(s), and a way to implement them? Thanks in advance.
Could someone tell me where it is that I might be going wrong with the following problem? Ok, so I'm computing a spectrogram, and, trying to return the PDS of the signal (to show where the signal has the most energies) which I will then use in plotting the decibels. In my approach, I multiply each of the frames by the ...
Hi I was researching on STFT when I encountered this slide online I don't understand the points, Can anyone explain them? Thanks in advance.
I'm trying to compute the real cepstral coefficients of recorded telephone audio in Matlab using the rceps function. On some audio frames (480 samples per frame -- 60ms of audio at 8kHz), I get a Matlab error: "rceps:ZeroInFFT". The cepstrum does not exist because some of the DFT coefficients are 0. The frames in ques...
I have done with iris localization. Now I need to perform iris normalization, but I don't have any idea about it. How to implement iris normalization please?
Ok, so I have some recorded data over several hours that looks like this (values on y, time on x). See this image http://imgur.com/fdakUlk I need a way to use matlab to divide the data into sections depending on their mean value. The sections needs to be atleas a couple of hours. Just by looking at the data they can ea...
I just started learning about image processing. I am using the book "Digital image processing" by Gonzalez and Woods but I don't like this book very much. I would like to read a book which can actually make me interested in image processing. Can you suggest some good resources?
According to my question about separability of Gabor filters in this link, I want now to convolve my image with this separable filter by using the normalized cross correlation operation. Assume my Gabor filter is G, my image is I. My Gabor is separated into Low-Pass gaussian filter f(x) and Band-Pass gaussian filter g(...
I am working on signal processing for speaker verification. I have extracted MFCC features for few wav files using GMM algorithm. Now, i want to add any 2 MFCC features to another variable. How can I achieve this?
What is the conceptual difference between Kolmogorov-Sinai entropy, Shannon entropy and Boltzmann entropy? Are they interchangeable and mean the same? Where can I find a good lucid explanation about Kolmogorov entropy and why it is needed and its significance. Thank you.
I'm trying to analyze the power spectral density of a signal recorded from a Ham radio antenna. The signal is about 30 seconds long, recorded at a 48 kHz sample rate. The frequency is center frequency of 8 kHz recording, with the receiver centered on 5.275 MHz. The signal I'm currently looking at is supposed to be arou...
I have several seconds of speech signal and I have to detect moments in time of arousal with excitation signal, which is creating by larynx. I'm looking for useful reference since several days. I know that a speech signal is a convolution of excitation signal with a filter response and I've read about cepstral analysis...
I have a doubt related to calculating the Discrete Time Fourier Transform (DTFT) by hand. Specifically in how calculate the frequency axis of the spectrum. My signal has N values and was sampled at FS Hz, the spectrum would have N entries too (where N/2 values are a mirror of the other half). The maximum representable ...
I want to extract the oilspill affected areas in SAR images using thresholding, where the affected area appears as darker than surroundings. Here I am attaching the code, where I am able to extract the white portions in the image. i=imread('oil.jpg'); figure(1); imshow(i) title('Orginal image'); % RGB to Gray convers...
I wanted to consult with some experts whether or not this pre-processing step is excessive for what I'm trying to achieve. I am in a similar scenario as the author of the article, where I have many many colonies that I'm trying to segment and distinguish from the background. So I found this article: A System for Automa...
How would you use the MATLAB function below to create a FIR filter with a custom magnitude and phase response? It is from the Statistical Modeling and Inference MATLAB Toolbox function [a, rw, info] = irls(X, y, wf, s, r2, a0, varargin) % Iterative Reweighted Least Square % % The iterative reweighted least square al...
I have to find the partial derivative of an image with respect to its x dimension. I am using central difference method i.e $ \partial_x F(x) = \frac{F(x+1,y)-F(x-1,y)}{2} $ Here $F(x,y)$ represents the image and if I want to use spatial filtering for the same then I can use filter mask as $0.5 \times [0, -1, 0;0, 0, ...
According to my question in this link, I applied (with a big help from lennon310) the separability of Gabor filters of any orientation. The only disadvantage of this method is the convolution in the complex domain. So we can treat the Gabor filter G(x,y) as a two dimensional matrix that has a low rank, since the filter ...
I'm calculating the beat of song through the frequency domain. I have 60 seconds of audio sampled at 48000Hz, multiplied by a Hanning window in the time domain. This gives me a resolution in the frequency domain of 1BPM, and I'm interested in frequencies from ~10 BPM to ~300BPM, which is only the first 300 bins of the ...
I know for sure that white noise is considered as a stationary sound, but is that true for the rest of them? http://en.wikipedia.org/wiki/Colors_of_noise Also, can we meet some of these sounds in nature? Is there any relation between these noises and noises that you can hear in a town(vehices,traffic,people chatting) o...
So far, I have read that all the non-parametric estimation techniques decrease the frequency resolution in order to decrease the variance in the spectral estimate What is the general "overview" meaning of variance and what is the advantage in decreasing the variance of spectral estimate? The PSD can be calculated from ...
I understand the cyclic prefix, but in (1) it's said that: "If we consider that the channel effect is minimal and can be neglected. then the phase difference of the CP and the OFDM symbol which the victim of CFO can be found by $2\pi\beta N/N= 2\pi\beta$." and then, the author derives the equation for estimating the CF...
I have got a video containing 200 frames and i am extracting low level features at pixel level. Then i am dividing the entire frame in to small spatial patches (say of size 7x7) and creating a feature vector for each spatial patch (by summing up the feature values of the pixel inside the particular patch). If i want to...
In certain applications, you have enough SNR available to, for example, perform an FFT and identify peak location and hence the signal frequency. If my understanding is correct, parameter estimation techniques, such as maximum likelihood, will only be useful when your SNR is so low that you can't run a basic peak searc...
I'm currently reading this paper: http://www.rssd.esa.int/SP/LISAPATHFINDER/docs/Data_Analysis/GH_FFT.pdf On page 21, it has a section called "Putting it all together", it states the following steps: Input data Frequency resolution and length of DFT Window function Splitting of the data stream FFT Scaling Now, the "W...
Ok, so finally some things are making sense. I'm trying to scale the results, which I have done by the following equation: $$ PS_{rms} = \frac{2 \cdot \left |y _{m} \right |^{2}}{ S_{1}^{2} } m = 0.....N/2 $$ Where $y_{m} = (re*re+im*im)$ This produces the following: (Sorry about the axis - I have not had time...
I'm trying to relate a near shore tidal signal (point A) to 3 points along a long model boundary (points B C D). I want to possibly have a relationship between B C D with which we can convert A predictions into B C and D. At the moment I'm doing a single phase shift, an amplitude ration for levels above zeros, an ampli...
When I was studying dispersion of refraction index in semiconductors and dielectrics, my professor tried to explain that if a filter (like a dielectric absorbing some light frequencies, or an electric RC-filter) removes some frequencies, then the remaining ones must be phase shifted to compensate for those frequencies ...
I would like to know what is the Compression Algorithm used by Facebook for compressing the images? I uploaded a image of size 242 Kb and I downloaded the same image from my Facebook album. The size of downloaded image is 46 Kb and the subjective quality is similar, I wasn't able to differentiate between the original (...
I have just a quick question about how to calculate Deltas from MFCC. I have found the formula for calculating deltas here: $$d_t=\frac{\sum_{n=1}^{N}n\left(c_{t+n}-c_{t-n}\right)}{2\sum_{n=1}^{N}n^2}$$ Another source states it is a regression equation. From what I can see, it takes coefficient from a previous and next...
I am pursuing a master's in signal processing and I am working on object tracking. Till now I have done object tracking using colors, background subtraction and the Lucas-Kanade method along with a Kalman filter. For my final dissertation, I want to do something new on object tracking and I have explored a lot to get s...
In QPSK, after splitting the odd and even bits into the I and Q channels, and then multiplying them with the sine and cosine function, what next am I suppose to do? Do I rearrange them back into their odd and even positions? Also, in OQPSK do I make the first bit of my Q array a dummy bit zero to represent the half s...
As a part of my thesis work -in which I am using both acoustic and visual information- I am looking forward to implement a real time sound source localization sub-system. From the geometric principles, I know that at least 4 non-coplanar microphones are needed for 3D localization. However, as I understand, using only a...
If I ave 2 otherwise identical 1D signals that are phase shifted from each other then cross-correlation is a perfect way to identify the time lag between the 2 signals. Now supposing I double the speed of one of those signals (eg sampled at every other sample). The cross correlation will fail miserably at this point....
Trying to understand whether or not I am calculating the PSD correctly. Below is the equation that I am using: $$ PSD_{rms}(f_{m} = m \cdot f_{res}) = \frac{PS_{rms} }{ENBW} = \frac{2 \cdot \left | y_{m} \right |^{2}}{f_{s}\cdot S_{2}} m = 0....N/2 $$ This equation is applied using the following: 1) Calculate the squar...
I have a 2D array/matrix (something like an image, but it's not an actual image) with values from 0 to X (in this case 180). and I have a 1D array/Vector that has the occurrence frequency of these values in the 2D matrix (I think it's like Histogram, but I've done that manually). so the 1D array can look like this. I ...
I am working recognition of plant seeds with Matlab, and I am looking for code to remove the shadow in grayscale and binary. The binarized version of the image is:
I am developing a chord recognition program using neural networks.I need to normalize the spectrogram or my pitch class profile to provide inputs for the neural network. The pitch class profile classifies the frequency into 12 class semitones (From note A to G). The problem here is it produces really high values and l...
I have a progam in matlab that generates a linear and quadratic chirp, and now I want to study the frequency components in wvtool. However, I am not able to decode the frquency domain scale "the normalized frequency". The usual process of deviding the normalized frequency with the sampling frequency doesnot provide the...
I'am pretty new to this topic so i dont have much experience with DSP. I want to filter(highpass) a WAVE file, i have already programmed the FFT, and now want to filter the FFT vector via convolution (which is a multiplication in frequency domain ?!). 1) Can anyone give me an easy equation for such a filter? 2) If i ha...
I understand the concept of channel capacity as the maximal rate of the channel code I can apply without making a mistake in the receiver, in that sense the capacity is between 0 and 1. What I don't understand is the meaning of the capacity in the AWGN channel case where it is calculated by C=(1/2)*log2(1+SNR) where cl...
Since a couple of months ago I started being actively involved in the area of control of dynamical systems. In most cases, designing a controller for a given dynamic system will require the employment of digital signal processing techniques, especially in the area of signal filters. Since I do not have a background in ...
My goal is to obtain a plot with the spatial frequencies of an image - kind of like doing a fourier transformation on it. I don't care about the position on the image of features with the frequency f (for instance); I'd just like to have a graphic which tells me how much of every frequency I have (the amplitude for a f...
Not sure if this is the right place for my question but let me give it a try. I have a very simple analog sensor (piezo transducer) which is attached to a microcontroller. I'm using this sensor to detect gestures (hand movements). The mCU sends the sensor data to my PC where I'm running a simple application that plots ...
Are all isotropic bandpass filter kernels, wavelets? Are there any properties that the radial frequency profile has to meet to be considered a wavelet?
I am trying to learn this part but I dont seem to understand. if I have a finite length sequence, and a n-point DFT of it (in an interval)... is it possible to evaluate expressions of the n-point DFT without computing them explicitly? the exercise is: Let $x[n]$ be finite-length sequence such that: $x[n] = \{2,1,1,0,...
I am working on an 802.11a demod that works, for the most part, but has a bug that causes intermittent errors. I haven't finished characterizing it, but it appears that the problem is in my equalizer block. 802.11a is an OFDM signal, and each OFDM symbol has 64 sub-channels. Four of those sub-channels are pilots (kn...
I have read somewhere online that the log gabor filter has an advantage over the gabor filter, in the sense that it has 0 DC component. How do you prove this property mathematically? Thanks in advance.
I want to implement an OFDM transceiver chip and my question is how to choose the clock frequency of the IFFT block? The number of output bits of the IFFT block? Also, how has the clock frequency and the number of output bits of the digital-to-analog converter (DAC) to be chosen?
How can i get lpc coefficient by htk? I know that there is a binary code "HCopy" which is ganna help with that... but my question is how can i read a wav file in htk? and i guess i need activeperl to run this? right? i installed htk but do i need compilation for getting lpc coefficient? I would appreciate if some one i...
I'm going to train a HOG detector for detect objects (pedestrian head) with SVMLight library. But my question is not about implementation. I'm wondering how to collect samples. I have to use some videos, and they have 1624x1234px dimension. The pedestrian heads have more or less 20x20px bounding-boxes. Is the 20x20 pix...
The well-known relationships for zero-mean circularly-symmetric complex Gaussian $z = a + jb = |z| \exp(j\varphi)$ signals are the amplitudes $|z| = \sqrt{a^2 + b^2}$ are Rayleigh-distributed the phases $\varphi = \arg(z)$ are uniformly distributed the pdf of such signals is fully described by their complex covariance...
Main Problem: How can you inverse Wavelet Transforms (Using the data given by signal.scipy.cwt) I was wondering if anyone understands the scipy.signal.cwt() function well enough to use it to remove backgrounds and noise from data. I have seen where Matlab has an inverse continuous wavelet transform function which will ...
What are the methods through which we can quantify the randomness or complexity in a given signal. I know spectral flatness measure (geometric to arithmetic means) is one way to do it, but what are other common techniques?
When looking for a sparse solution to $\ Az = y $, under what condition(s) does the nullspace $\ N(A)$ of a matrix $\ A$ not contain any $\ 2s$-sparse vector other than the zero vector, i.e $\ N(A) \cap \{z \in \mathbb{R}^N: \|z\|_0 \le 2s\} = \{0\} $
When designing digital filters we start off with FIR and IIR. Later on we talk about window functions e.g Kaiser and Hamming windows. I do not understand why we need so many type of window functions described by rather complex mathematical equations and how they relate to FIR and IIR.
i have to do super resolution of two low resolution images to obtain a high resolution image. 1st image is taken as base image and the second image is registered with respect to it . i used SURF algorithm for image registration . A Delaunay triangulation is constructed over the points using a built-in MATLAB delaunay f...
I have the following Spectrogram of a signal: Essentially, in this signal, I know there is one point of interest (between 60-80) and between the the particular frequencies. I could, potentially smooth the signal and perform a simple peakfinding algorithm to find the peak and then just extract the bin number where the...
Why do people sometimes write $H(e^{j\omega})$, some others use $H(\omega)$ and even others use $H(f)$ with $\omega=2\pi f$ to describe the frequency response of a filter or a spectrum? Is there any "real" difference in meaning or is it just convention?
Fore-word :Kindly Observe this question has multiple updates take the latest update as question ! here below is my 256pt FFT on an 2Khz sinusoid signal,on a fixed point DSP tms320c5515 please ignore this this has occured due to programatic bug is this above figure expected in the first place ?(with out windowing) afte...
This is a follow up question to: Segment/Idenfication of signal from a spectrogram My main goal is to identify (segment) all of the calls from a given signal. The signal looks like the following (Spectrogram): Notice how there are 4 calls within this signal, I would like to identify there are 4 calls and just capture ...
In (1), it's said that: If two identical training symbols are transmitted consecutively, the corresponding signals with CFO of $\epsilon$ are related with each other as follows: $y_2[n]=y_1[n]\exp(j2\pi \epsilon)$. How is this relation between $y_1[n]$ and $y_2[n]$ achieved? How is the exponential part associated wi...
I'm trying to convert accelerometer data from accelerations to displacement, I assume that the initial velocity is zero and the initial position is also zero. Here's my code, Fs = 500; T = 1/Fs; t = (0:length(y) -1)*T; vel = cumtrapz(t, acc); disp = cumtrapz(t, vel); here are the respective plots from acceleration to...
Main Question: Why would iterative wavelet/inverse-wavelet transforms cause a shift along the x-axis for undecimated (shift-invariant) wavelet transforms? I am attempting to remove backgrounds from signals using an iterative wavelet transform method similar to this approach which I found in an article: However, I am r...
My $G(s)=1-e^{-s/\tau}$, $\tau$ is very small, say of order $10^{-4}$. I need to compute a $H(z)$ (a digital filter) such that $H(z)$ has the inverse response of $G(s)$. Is ok even if $G(z)$ has the inverse response just for $f$<20k. I don't not also zero and pole go $G(s)$. I'm able to have an proximation of $H(z)$, b...
My question is simple but I am new to MATLAB's sound functions so I am unsure what to do. I would like to read the audio (from the microphone) then apply the fft(X,n) function to it where the n should be 256. I was applied the fft function to this code like fft(myRecording,256) but I kept getting zeros.
I have a beam fixed at one end and free at the other. It obviously results in vibrations due to the disturbances in the environment or when induced. This system can be represented using a differential equation. The vibrations are sensed using sensors, analyzed and reduced. Now, the sensing is done through compressed se...
I would like to create a signal that is nearly uniform distributed (as depicted in Figure 1). It must consist of some single sine waves (e. g. 400 Hz, 430 Hz, 500 Hz) or a random band limited signal (e g. 495 to 500 Hz as depicted in Figure 2). I expiremented with some parameters but I didn't get a good result. Does an...
I am collection a quadrature signal from a radio. The I and Q signals go through a sound card at a sample rate of 192000, and then a FFT is performed on the signals. Due to various reasons, the signals are shifted slightly from the 90 degree phase difference and need to be corrected at some point. How can one go about...
I am looking for a good introduction to wavelets and wavelet transforms. that covers the following: Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and Orthonormality – Relationship Between Vectors and Signals – Signal Spaces – Concept of Convergence – Hilbert Spaces for Energy Signals- Fouri...
How is the filter() implemented in MATLAB? How is it so fast? What's the fastest implementation of a FIR filter?
Suppose I have the following signal (Spectrogram): I have three types of elements that I want to identify within this signal: 1) Noise 2) Call-1 3) Call-2 I have prior training of what each of the elements looks like and can obtain training for the duration of each call, and, the frequency ranges that each call will...
I am an ECE student, and i am doing my project I want to generate a code for gabor wavelet transform. I have no idea could you help me with a sample program I have DNA as input sequences and how can use it as input data "sample sequence s={ACGTACGTACCCCAGGGATTT} " thanks and regards!
$y(n)=n \times y(n-1)+x(n)$ $y(n)=$ output of causal Linear Time Invariant system $x(n)=$ input of causal Linear Time Invariant system $y(n)=x(n) * h(n)$ $h(n)= $system
The figure below shows in dashed lines sinusoidal signals of the same frequency at three different phase shifts. The signals are then sampled such that the sinusoidal frequency is exactly a half of the sampling frequency, i.e. the frequency of all the sinusoids is the Nyquist frequency. The samples taken from this si...
This is my system. Is there a way to find a transfer function? I can’t help the YR(s) going inside B(s) without being sampled, and I think this is the problem. Any help?
I have a collection of audio files of subjects sleeping, which have different lengths, ranging from 8 hours to 10 hours. I want to detect when a snore occurs in them. First of all I want to normalise them. What is the best method to normalise them so they all have similar values. Currently I'm am using the followi...
I am trying to solve a phasor addition problem, reducing from its original form to $$X(t) = A \cos(\omega_0 * t + \phi)$$ The original equation is : $$X(t) = 2 \sin(\omega_0 * t + 45) + \cos(\omega_0*t)$$ I converted the sine term into cosine. The result that i got was $$X(t) = 2 \cos(\pi/2 - \omega_0*t - \pi/4) + \...
I used nftool in Matlab 2012 and trained a network. I gave the training inputs as x=[250:1] and targets as t=[250:1]. I used 10 hidden layers. I trained the network and got the results and exported the data to workspace. I finally got net as the variable in the workspace. I want to realize this network in hardware. So,...
Does anyone know any FFT algorithm that can be implemented in C? I am trying to convert a built-in matlab FFT api to a function in C. Appreciate any help!
Right. I'm trying to understand the concepts for determining where the signal contains voiced and unvoiced parts. I have created a spectrogram (STFT) of a signal and I'm trying to separate from voiced and unvoiced. I have tried calculating this using the total energy for each bin, and, applying some threshold value f...
When taking the DTFT of a discrete-time signal $x[n]=a \times u[n]$, the book "Signals and Systems" by Chi-Tsong Chen claims that the result is $\frac 1 {1-a \cdot e^{-j \cdot \omega \cdot T}}$; however, many other sources claim the DTFT is $\frac 1 {1-a \cdot e^{-j \cdot \omega}}$ without the T term. The book I have s...