Lecture 7 - The Discrete Fourier Transform
Figure 7.2: Example signal for DFT. Let us sample вдгжеиз at 4 times per in order to calculate the DFT. The solution is to use one of the window ...
ABC of DFT: Hands-on session 1 Introduction into calculations on
9 нояб. 2012 г. Example 1 – H atom: running calculation. To run a DFT calculation call a dscf module: dscf. Scroll up output data: you'll find
6 A Tutorial on Density Functional Theory
In particular we do not include any explicit exam- ple of a localized basis set DFT calculation. The final example
Running a DFT calculation in VASP
For very accurate total energy calculations (no relaxations in metals) use the tetrahedron method (ISMEAR=-5). • k- point convergence. – recommended to use even
ABC of DFT: Hands-on session 2 Molecules: structure optimization
16 нояб. 2012 г. Example 1 – Benzene: running calculation. To run a DFT calculation call from the current directory ridft > ridft.out &. After a minute ...
New DFT+Hubbard input in Quantum ESPRESSO (since v.7.1)
30 мар. 2023 г. Below is the example showing how to perform DFT+U+J0 calculation: &control calculation='scf' restart_mode='from_scratch' prefix='Ni2MnGa ...
Density Functional Theory (DFT)
fi). The algorithm is surprisingly simple to implement as one needs only few lines of code. Here is the example for u = 0 (usual Schroedinger equation):.
Lecture 7 The self-consistent field procedure for Kohn-Sham DFT
CHEM6085 Density Functional Theory. Example: DFT calculation on a water molecule. SCF procedure convergence iter energy DeltaE. RMS-Dens Diis-err time.
- An ab initio DFT and semiempirical SCF-MO package - Version
example let us calculate a number for the oxygen-oygen dissociation energy in the H2O2 ... calculation: --------------------------. THERMOCHEMISTRY AT 298.15K.
Lecture 7 - The Discrete Fourier Transform
We could regard each sample ??? &' as an impulse having area ???R&S . DFT equation for the fundamental frequency (one cycle per sequence `Xc Hz
Density Functional Theory
Example: electronic structure of benzene (42 electrons) Historically: modify existing codes for HF calculations to perform DFT calculations.
Untitled
18 ??.?. 2562 properties were calculated by using density functional theory. (DFT) and the time-dependent density functional theory (TDDFT).
Lecture 7 The self-consistent field procedure for Kohn-Sham DFT
CHEM6085 Density Functional Theory. Example: DFT calculation on a water molecule. SCF procedure convergence iter energy DeltaE. RMS-Dens Diis-err time.
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nanomaterials by using the quantum chemistry calculations under the density functional theory has been presented. Structural properties molecular orbitals
ABC of DFT: Hands-on session 1 Introduction into calculations on
9 ?.?. 2555 ABC of DFT Hands-on session 1: Introduction into calculations on molecules. Example 1 – H atom. We start with the simplest possible system: ...
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Adsorption energies were calculated for organosulfur (thiophene) DFT ???????????????????????????????????????????? ( )n r ???????? Normalized ? ???.
6 A Tutorial on Density Functional Theory
6.4 we present our first test case namely atoms
Chapter 5 - The Discrete Fourier Transform
Example. Find the 8-point DFT of the signal x[n] = 6 cos2(? can recover it from xps[n] which in turn can be recovered from the DTFS synthesis equation.
Discrete Fourier Transform (DFT)Recall the DTFT:
X(ω) =∞?
n=-∞x(n)e-jωn.DTFT is not suitable for DSP applications because•In DSP, we are able to compute the spectrum only at specific
discrete values ofω,•Any signal in any DSP application can be measured only in a finite number of points.A finite signal measured atNpoints:
x(n) =? ?0, n <0,0, n≥N,
wherey(n)are the measurements taken atNpoints.EE 524, Fall 2004, # 51Sample the spectrumX(ω)in frequency so that
X(k) =X(kΔω),Δω=2πN=?
X(k) =N-1?
n=0x(n)e-j2πknNDFT.Theinverse DFTis given by:
x(n) =1NN-1? k=0X(k)ej2πknN. x(n) =1NN-1? k=0? N-1? m=0x(m)e-j2πkmN? e j2πknN N-1? m=0x(m)?1NN-1?
k=0e -j2πk(m-n)N?δ(m-n)=x(n).EE 524, Fall 2004, # 52
The DFT pair:X(k) =N-1?
n=0x(n)e-j2πknNanalysis x(n) =1NN-1? k=0X(k)ej2πknNsynthesis.Alternative formulation:X(k) =N-1?
n=0x(n)Wkn←-W=e-j2πN x(n) =1NN-1? k=0X(k)W-kn.EE 524, Fall 2004, # 53EE 524, Fall 2004, # 54
Periodicity of DFT SpectrumX(k+N) =N-1?
n=0x(n)e-j2π(k+N)nN N-1? n=0x(n)e-j2πknN? e -j2πn =X(k)e-j2πn=X(k) =? the DFT spectrum is periodic with periodN(which is expected, since the DTFT spectrum is periodic as well, but with period2π).
Example:DFT of a rectangular pulse:
0,otherwise.
X(k) =N-1?
n=0e -j2πknN=Nδ(k) =? the rectangular pulse is "interpreted" by the DFT as a spectral line at frequencyω= 0.EE 524, Fall 2004, # 55 DFT and DTFT of a rectangular pulse (N=5)EE 524, Fall 2004, # 56 Zero PaddingWhat happens with the DFT of this rectangular pulse if we increaseNbyzero padding: {y(n)}={x(0),...,x(M-1),0,0,...,0????N-Mpositions},
wherex(0) =···=x(M-1) = 1. Hence, DFT isY(k) =N-1?
n=0y(n)e-j2πknN=M-1? n=0y(n)e-j2πknN sin(πkMN)sin(πkN)e-jπk(M-1)N.EE 524, Fall 2004, # 57DFT and DTFT of a Rectangular Pulse with
Zero Padding (N= 10, M= 5)Remarks:•Zero padding of analyzed sequence results in "approximating" its DTFT better,•Zero padding cannot improve the resolution of spectral components, because the resolution is "proportional" to1/Mrather than1/N,•Zero padding is very important for fast DFT implementation
(FFT).EE 524, Fall 2004, # 58 Matrix Formulation of DFTIntroduce theN×1vectors x=? ??x(0) x(1)... x(N-1)? ??,X=? ??X(0)X(1)...
X(N-1)?
and theN×Nmatrix W=? ?????W0W0W0···W0
W0W1W2···WN-1
W0W2W4···W2(N-1)
W0WN-1W2(N-1)···W(N-1)2?
DFT in a matrix form:
X=Wx.Result:Inverse DFT is given by
x=1NWHX,EE 524, Fall 2004, # 59 which follows easily by checkingWHW=WWH=NI, where Idenotes the identity matrix. Hermitian transpose: xH= (xT)?= [x(1)?,x(2)?,...,x(N)?].
Also, "
?" denotes complex conjugation. Frequency Interval/Resolution:DFT"s frequency resolution F res≂1NT[Hz] and covered frequency intervalΔF=NΔFres=1T=Fs[Hz].
Frequency resolution is determined only by the length of the observation interval, whereas the frequency interval isdetermined by the length of sampling interval. Thus•Increase sampling rate=?expand frequency interval,•Increase observation time=?improve frequency resolution.
Question:Does zero padding alter the frequency resolution?EE 524, Fall 2004, # 510 Answer:No, because resolution is determined by the length of observation interval, and zero padding does not increase this length. Example (DFT Resolution):Two complex exponentials with two close frequenciesF1= 10Hz andF2= 12Hz sampled with the sampling intervalT= 0.02seconds. Consider various data lengthsN= 10,15,30,100with zero padding to 512 points.DFT withN= 10and zero padding to 512 points. Not resolved:F2-F1= 2Hz<1/(NT) = 5Hz.EE 524, Fall 2004, # 511DFT withN= 15and zero padding to 512 points.
Not resolved:F2-F1= 2Hz<1/(NT)≈
3.3Hz.DFT withN= 30and zero padding to 512 points.
Resolved:F2-F1= 2Hz>1/(NT)≈1.7Hz.EE 524, Fall 2004, # 512DFT withN= 100and zero padding to 512
points. Resolved:F2-F1= 2Hz>1/(NT) =0.5Hz.EE 524, Fall 2004, # 513
DFT Interpretation Using
Discrete Fourier SeriesConstruct a periodic sequenceby periodic repetition ofx(n) everyNsamples: {?x(n)}={...,x(0),...,x(N-1)???? {x(n)},x(0),...,x(N-1)???? {x(n)},...} The discrete version of the Fourier Series can be written as ?x(n) =? kX kej2πknN=1N? k?X(k)ej2πknN=1N?
k?X(k)W-kn,
where ?X(k) =NXk. Note that, for integer values ofm, we have W As a result, the summation in the Discrete Fourier Series (DFS) should contain onlyNterms: ?x(n) =1NN-1? k=0?X(k)ej2πknNDFS.EE 524, Fall 2004, # 514
Inverse DFS
The DFS coefficients are given by
X(k) =N-1?
n=0?x(n)e-j2πknNinverse DFS.Proof. N-1? n=0?x(n)e-j2πknN=N-1? n=0? ?1NN-1? p=0?X(p)ej2πpnN?
e-j2πknN N-1? p=0? X(p)?1NN-1?
n=0e j2π(p-k)nN?δ(p-k)=
?X(k).The DFS coefficients are given by
X(k) =N-1?
n=0?x(n)e-j2πknNanalysis, ?x(n) =1NN-1? k=0?X(k)ej2πknNsynthesis.EE 524, Fall 2004, # 515
•DFS and DFT pairs are identical, except that-DFT is applied to finite sequencex(n),-DFS is applied to periodic sequence?x(n).•Conventional (continuous-time) FS vs. DFS-CFS represents a continuous periodic signal using an
infinite number of complex exponentials, whereas-DFS represents a discrete periodic signal using a finite number of complex exponentials.EE 524, Fall 2004, # 516DFT: PropertiesLinearity
Circular shift of a sequence: ifX(k) =DFT {x(n)}thenX(k)e-j2πkmN=DFT {x((n-m)modN)}
Also ifx(n) =DFT-1{X(k)}then
x((n-m)modN) =DFT-1{X(k)e-j2πkmN} where the operationmodNdenotes the periodic extension ?x(n)of the signalx(n): ?x(n) =x(nmodN).EE 524, Fall 2004, # 517DFT: Circular ShiftN-1?
n=0x((n-m)modN)Wkn =WkmN-1? n=0x((n-m)modN)Wk(n-m)EE 524, Fall 2004, # 518 =WkmN-1? n=0x((n-m)modN)Wk(n-m)modN =WkmX(k), where we use the facts thatWk(lmodN)=Wkland that the order of summation in DFT does not change its result.Similarly, ifX(k) =DFT {x(n)}, then
X((k-m)modN) =DFT {x(n)ej2πmnN}.
DFT: Parseval"s Theorem
N-1? n=0x(n)y?(n) =1NN-1? k=0X(k)Y?(k) Using the matrix formulation of the DFT, we obtain yHx=?1NWHY?
H?1NWHY?
1N2YHWWH????
NIX=1NYHX.EE 524, Fall 2004, # 519
DFT: Circular Convolution
IfX(k) =DFT {x(n)}andY(k) =DFT {y(n)}, then
X(k)Y(k) =DFT {{x(n)}?{y(n)}}
Here,?stands for circular convolution defined by
{x(n)}?{y(n)}=N-1? m=0x(m)y((n-m)modN).DFT {{x(n)}?{y(n)}}
N-1? n=0? ?N-1 m=0x(m)y((n-m)modN)? {x(n)}?{y(n)}W kn N-1? m=0? ?N-1 n=0y((n-m)modN)Wkn?Y(k)Wkmx(m)
=Y(k)N-1? m=0x(m)WkmX(k)=X(k)Y(k).EE 524, Fall 2004, # 520
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