Questions tagged [dtft]
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171 questions
0 votes
0 answers
73 views
Orthogonality of Finite Samples of a Function and Its Delayed Version
Consider a band limited function $f(x)$ in $L_2(\mathbb{R})$ with frequency support in $[-B,B]$ . For small delays $\delta$ it is straightforward to show that $f(x)$ and $f(x+\delta)$ are not ...
2 votes
0 answers
83 views
Why are books on dsp so bothered about mean square convergence for DTFT but not for CTFT?
I was going through dsp book authored by Proakis. I’m a bit confused to see that there is a weak condition for DTFT which is the mean square convergence applicable for finite-energy signals which are ...
- 131
2 votes
1 answer
221 views
Dimensionally consistent discrete and continuous Fourier Transforms
If a signal $x(t)$ is sampled $x[n] = x(n\Delta t)$, then the discrete Fourier transform $X[k]$ of $x[n]$ should approximate the continuous Fourier transform $X(f)$ of $x(t)$ up to linear rescaling. [...
- 776
2 votes
2 answers
175 views
Show $\frac{B}{A}=-\frac{C_3}{C_2} \cdot \frac{3 \sqrt{3}}{4}$ for two pulse‐train Fourier coefficients
Question Consider the two periodic signals $$ \require{cancel} \xcancel{y(t)=B\sum_{n\in\mathbb{Z}}\operatorname{rect}(2t-n),\quad x(t)=A\sum_{n\in\mathbb{Z}}\operatorname{rect}\left(\tfrac{t}{2}-6n\...
- 75
2 votes
3 answers
2k views
Explanation of an Irregular Webcomic DSP joke
Irregular Webcomic! #1640 is a parody of an xkcd comic. I heard of this comic from the 2019-02-01 recording of UC Berkeley EE123 class, but it didn't give a detailed explanation. It was after the ...
- 23
2 votes
4 answers
473 views
Condition to create "false peaks" in the DTFT
The Dirichlet Kernel is the periodic frequency response of a rectangular window in time. The Discrete Time Fourier Transform (DTFT) of a rectangularly windowed signal is the convolution of its ...
- 58.2k
6 votes
4 answers
1k views
Will the peak of the DFT always be closest to the peak of the DTFT?
The Discrete Time Fourier Transform (DTFT) is a continuous function in the frequency domain, providing the magnitude and phase of a frequency response at every frequency in the unique span of DC to ...
- 58.2k
0 votes
0 answers
57 views
interpolation based filter structure frequency response
given the following structure: and given $$H_0(\omega) = \text{DTFT}(h_0[n])= \begin{cases} 1 & |w| \le \pi/M \\ 0 & \text{otherwise} \end{cases}$$ $h_k[n] = h_0[n] e^{\frac{j 2 \pi k n}{M}}, ...
1 vote
2 answers
89 views
Deriving DFT Scaling via DTFT and Convolution with Window Function
I think I've now read through all of the questions on DFT scaling on this site, and I've still got one more. (I'm sure the knowledge is here somewhere, but I couldn't put together the pieces!) I'm ...
- 313
1 vote
1 answer
154 views
DTFT of an autocorrelation function is producing a complex PSD?
I'm trying to find the PSD of a pretty simple autocorrelation function for a discrete random process, $R_x[k]=\cos(ω_0k)$. From a data-book for Z-transforms, $g_k =\cos(ω_0k)$ transforms into $$G(z)=\...
- 13
1 vote
1 answer
80 views
Why is the digital frequency response taken on the unit circle, while the analog is taken along the imaginary axis?
For digital signals, the fourier transform is taken along the unit circle of the Z-transform. The equivalent to the Z-transform in continuous signals is the Laplace transform, but in that case the ...
- 13
0 votes
1 answer
80 views
Computing the DFT of three sine waves with Aliasing
I was asked to compute the DFT of the following: $$x(t) = \sin(2 \pi 1000 t) + \sin(2 \pi 3500 t) + \sin(2 \pi 19000 t)$$ Sampled at $f_s = 20,000 [Hz]$ for $N=256$ samples. can you please look at my ...
0 votes
1 answer
64 views
Discrete-Fourier transform of $$u[-n+2]$$
I'm Trying to Find the fourier transform in discrete time for $$u[-n+2]$$ . My steps : Time-Reversal Property : $$ u[(-n+2)] \{\omega\} = u[-(-n+2)] \{-\omega\} = u[n-2] \{-\omega\} $$ Time-Shifting ...
- 75
1 vote
1 answer
145 views
Showing analytically that sampling exactly 1 period of a sinusoid yields a spectrum with no lobes in the DFT
The DTFT of a discrete sinusoid $f[k] = \sin(\omega_0 k)$ is $$F(\Omega)=i\pi(\delta[\Omega-\omega_0] + \delta[\Omega+\omega_0]), \: \: \: \Omega \in [-\pi, \pi)$$ The DTFT of a rect function $w[k] = ...
- 546
1 vote
1 answer
123 views
I want to understand the fundamental difference/connection between DFS, DFT and DTFT
I'm an EE student and I seem to miss some basic concept of my Signals course. We have learned about all the different Fourier methods available, but I don't seem to find a difference/understand it. As ...
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