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 \Delta \theta(k) &= 2\pi \frac{k}{N} n_0 \\ Therefore, substituting both the length of the sequence $L$ and observation window $N$ equal to $N/M$ in Eq \eqref{eqIntroductionDFTrectangleM} and doing the same for $Q$ part, we get\begin{aligned} But if the sinusoid in a rectangular signal is real with frequency $0$, why is the peak value in the magnitude plot equal to $N = 15$ rather than $N/2 = 7.5$? For understanding what follows, we need to refer to the Discrete Fourier Transform (DFT) and the effect of time shift in frequency domain first. . In that case, the peak of the mainlobe was equal to $L$, the length of the sequence. Symmetry property. shown in Figure below, where $N \ge L$. S_I[k] &= \frac{1}{\sin \theta/2} \cos \left(n_s + \frac{L-1}{2}\right)\theta \cdot \sin\left(\frac{L}{2} \right)\theta \\ \begin{align*} Thus, plugging $n_0=+1$ in the expression $2\pi (k/N) n_0$ gives the phase rotation $\theta (k)$ for each frequency bin as \sum \limits _{n=0} ^{N-1} \sin 0 &= 0 \begin{align*} $X(e^{j\omega}) = \frac{1}{1-\frac{1}{4}e-j\omega} = \frac{1}{1-0.25\cos \omega+j0.25\sin \omega}$, $\Longleftrightarrow X^*(e^{j\omega}) = \frac{1}{1-0.25\cos \omega-j0.25\sin \omega}$, Calculating, $X(e^{j\omega}).X^*(e^{j\omega})$, $= \frac{1}{(1-0.25\cos \omega)^2+(0.25\sin \omega)^2} = \frac{1}{1.0625-0.5\cos \omega}$, $\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{1.0625-0.5\cos \omega}d\omega$, $\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{1.0625-0.5\cos \omega}d\omega = 16/15$, We can see that, LHS = RHS.\end{align*} &\cdots \\ \begin{align*} Sometimes it is easy to get confused from the fact that $Q$ component of $s[n]$ is $0$ in the above example, leading to an incorrect phase result of $0$ or $\pi$. The DC, or average, value of this signal $\frac{1}{N}\sum s[n]$ is $0$ which is evident from Eq \eqref{eqIntroductionDFTPhaseExample} as the average value of a sinusoid over an integral number of cycles is $0$. Then, Considering the fact that $s[n]$ is composed of two real sine waves with amplitudes $1$ and $0.75$, we can see that $1$ kHz sine wave has a magnitude $1\times N/2 = 4$ and $2$ kHz sine wave has a magnitude $0.75 \times N/2 = 3$.Since the starting sample $n_s = 0$, the angle is equal to $0$. Note that the above equation is very similar to Eq \eqref{eqIntroductionDerivation1} encountered while computing the DFT of a rectangular sequence. F &= F_S \cdot \frac{k}{N} \\ The peak amplitude of the mainlobe is $L$, the output magnitude of the DFT for that particular sinusoid is $AN$, the output magnitude of the DFT for a real sinusoid is $AN/2$. \Delta \theta(\pm3) &= 2\pi \frac{\pm3}{N} \times \frac{180^\circ}{\pi} = \pm 67.5^\circ \\ \Delta \theta(\pm2) &= 2\pi \frac{\pm2}{N} \times \frac{180^\circ}{\pi} = \pm 45 ^\circ \\ \end{align*} Here, we discuss a few examples of DFTs of some basic signals that will help not only understand the Fourier transform but will also be useful in comprehending concepts discussed further. Below, Theorem 1 and Example 1 offer a more precise description as to why this is. DIT … \end{align*} The shape of the sinc function did not change as compared to a $4$ kHz input but it is sampled in frequency domain at non-ideal points — fractional frequency bins due to non-integer number of cycles of the input in time domain. And if the input signal contains contributions from only these frequencies, the DFT output is proportional to the magnitude at the contributing frequency bins and zero at the remaining bins. The $Q$ component here is $0$ in time domain only but the phase plot is determined by $IQ$ components in frequency domain. For time traveling in the past (i.e., a signal delay), the phase rotation is also negative while for time traveling in the future (i.e., a signal advance), the phase rotation is also positive. \begin{align*} Therefore, Here, the spectral replicas have a peak magnitude of $5$ instead of $15$ which suggests that the actual spectrum has been scaled down by a factor of $1/M=1/3$. s_I[n] = \left\{ \begin{array}{l} Let samples be denoted . The input/output relationship in frequency domain is: Substituting, m = (n/L) Example: Commonly used General Properties of the DFT . Séquence d'appel. This is known as DFT leakage. This is because the DFT of a regular real sinusoid is two impulses with amplitude $N/2$, one at that frequency and the other at its negative counterpart, but at frequency $0$, both of these impulses merge together to give a magnitude of $N$. X = fft (A [, sign] [, option]) X = fft (A, sign, selection [, option]) X = fft (A, sign, dims, incr [, option] ) Arguments A. un tableau de nombres réels ou complexes (vecteur, matrice, ou tableau N-dimensionnel). See illustration on subsequent page. Similarly, the DFT of a real sinusoid $\cos[2\pi (k’/N)n]$ can be seen as the sum of the above expression with a similar expression replacing $k-k’$ with $k+k’$ because each real sinusoid consists of two complex sinusoids scaled by half. In this case, fft pads the input sequence with zeros if it is shorter than n, or truncates the sequence if it is longer than n. If n is not specified, it defaults to the length of the input sequence. \Delta \theta(0) &= 2\pi \frac{0}{N} \times \frac{180^\circ}{\pi} = 0 \\ The bin $1$ frequency at $8000 \cdot 1/8 = 1$ kHz has a phase of $-90 ^\circ$. \sum \limits _{n=0} ^{N-1} 0 \cdot \cos 2\pi \frac{k}{N}n – \cos 2\pi \frac{k}{N}n \cdot \sin 2\pi \frac{k}{N}n &= 0 However, the process of calculating DFT is quite complex. A similar argument holds for the other frequency of $2$ kHz.\begin{equation*} \measuredangle S[k] = -2\pi\frac{k}{N} \left(n_s + \frac{L-1}{2}\right) \label{eqIntroductionDFTrectangleP} This leakage causes any input signal whose frequency is not exactly at a DFT bin center to leak into all of the other DFT output bins. In the DFT of a rectangular signal, we saw that both the numerator and denominator become equal to zero for $k=0$. Through the phase plot, the DFT in fact finds the time alignments of all the sinusoids at bin frequencies $F_S \cdot k/N$. 1, \quad n_s \le n \le n_s+L-1 \\ To complete the picture, we need one extra piece of information that we prove later: the DFT of a real and even symmetric signal is purely real with $0$ $Q$ part. There, a real sinusoid is a sum of two complex sinusoids and the phase of those two complex sinusoids in frequency $IQ$-plane determines the starting sample of the real sinusoid in time domain. From magnitude plot of this figure, observe that the DFT has detected two real sinusoids in this signal because the impulses at bins $1$ and $-1$ indicate the presence of two complex sinusoids that combine to form one real sinusoid at a frequency $8000\cdot1/8 = 1$ kHz. Verify Parseval’s theorem of the sequence x(n)=1n4u(n) Solution − ∑−∞∞|x1(n)|2=12π∫−ππ|X1(ejω)|2dω L.H.S ∑−∞∞|x1(n)|2 =∑−∞∞x(n)x∗(n) =∑−∞∞(14)2nu(n)=11−116=1615 R.H.S. k = \pm3 &\Rightarrow 8000 \cdot\frac{\pm3}{8} = \pm3~ \text{kHz} \\ The "N" is DFT is understood to be the number of data points in a given sequence or in other words the length of the sequence.The solid blue curve is shown in Figure above in time domain along with one analysis frequency $F_4 = F_S \cdot k/N = 16000 \cdot 4/16 = 4$ kHz. Part (b) of above Figure shows the phase plot for a right shift of $1$, i.e., $n_s = -(L-1)/2+1$. Not resolved: F 2 −F 1 = 2 Hz < 1/(NT) = 5 Hz. One final question: an input sinusoid at $3.7$ kHz sampling the sinc function at unaligned points in frequency domain makes sense, but why the graph in the earlier figure exhibits a similar kind of leakage? Is it a cosine with a phase of $-90^\circ$? Consequently, the peak value is seen to be $L = 16$ and $L = 7$ in their respective figures.\begin{align*} P_I[k] &= \sum \limits _{m=0} ^{N/M-1} \cos 2\pi\frac{k}{N} M m = \sum \limits _{m=0} ^{N/M-1} \cos 2\pi\frac{k}{N/M} m All applications of the DFT depend crucially on the availability of a fast algorithm to compute discrete Fourier transforms and their inverses, a fast Fourier transform. It is due to this integer multiplicity that DFT is able to find each $k-th$ contribution independently of all others. (Hence Proved), Compute the N-point DFT of $x(n) = 3\delta (n)$, $X(K) = \displaystyle\sum\limits_{n = 0}^{N-1}x(n)e^{\frac{j2\Pi kn}{N}}$, $= \displaystyle\sum\limits_{n = 0}^{N-1}3\delta(n)e^{\frac{j2\Pi kn}{N}}$, Compute the N-point DFT of $x(n) = 7(n-n_0)$, $\displaystyle\sum\limits_{n = 0}^{N-1}7\delta (n-n_0)e^{-\frac{j2\Pi kn}{N}}$, $\displaystyle\sum\limits_{-\infty}^\infty|x_1(n)|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X_1(e^{j\omega})|^2d\omega$, $\displaystyle\sum\limits_{-\infty}^\infty|x_1(n)|^2$, $X(e^{j\omega}) = \frac{1}{1-\frac{1}{4}e-j\omega} = \frac{1}{1-0.25\cos \omega+j0.25\sin \omega}$.\begin{align*} As discussed before, an N-point DFT and inverse DFT can be implemented as matrix multiplications where is the N by N DFT matrix with its mnth element being Consider the following cases for N=2, 4 and 8. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. Let us split X(k) into even and odd numbered samples. (1) we evaluate Eq.\begin{align} Example (DFT Resolution): Two complex exponentials with two close frequencies F 1 = 10 Hz and F 2 = 12 Hz sampled with the sampling interval T = 0.02 seconds.\begin{align*} Verify Parseval’s theorem of the sequence $x(n) = \frac{1^n}{4}u(n)$, Solution − $\displaystyle\sum\limits_{-\infty}^\infty|x_1(n)|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X_1(e^{j\omega})|^2d\omega$, L.H.S $\displaystyle\sum\limits_{-\infty}^\infty|x_1(n)|^2$, $= \displaystyle\sum\limits_{-\infty}^{\infty}x(n)x^*(n)$, $= \displaystyle\sum\limits_{-\infty}^\infty(\frac{1}{4})^{2n}u(n) = \frac{1}{1-\frac{1}{16}} = \frac{16}{15}$, R.H.S. Example 3: Consider an L up-sampler described by the discrete sequence . k = \pm2 &\Rightarrow 8000 \cdot\frac{\pm2}{8} = \pm2~ \text{kHz} \\ sequence • As in the 1D case, 2D-DFT, though a self-consistent transform, can be considered as a mean of calculating the transform of a 2D sampled signal defined over a discrete grid. Where did this number come from? With the original DFT input being exactly integer k cycles of a cosine sequence, to verify Eq. Referring to the phase plot and going back to the DFT definition, assume that the input is $\cos 2\pi (k/N) n$. Now consider the case of $L = 7$, $N = 16$ and $n_s = -(L-1)/2-1 = -4$. Given that the sequence is real valued with 8 points. Since $\sin \pm \pi = 0$, the first zero crossing occurs when the numerator argument in Eq \eqref{eqIntroductionDFTrectangleM}, $\sin \pi L k/N$, is equal to $\pi$. Using orthogonality, the DFT output for a complex sinusoid is given as, \begin{align*} To compute its DFT, consider that $p_I[n] = p[n]$ and $p_Q[n]=0$. That being the case, the sinc function becomes invisible and looks like two sets of impulses only.\end{align}. Next: Sampling Theorem Up: Discrete Fourier transform (DFT) Previous: Physical Interpretation of DFT DFT Examples. \label{eqIntroductionDFTgeneralSinusoid}. The width of the mainlobe is defined by zero crossings of the curve. Let () ≜ () be the sample sequence of a waveform () and s = 1/ be the sample rate. s[n] = \sin 2\pi \frac{4}{16} n \measuredangle S[k] &= -2\pi\frac{k-k’}{N} \left(\frac{N-1}{2}\right) Consider various data lengths N = 10,15,30,100 with zero padding to 512 points.\begin{align*} Then, the phase plot would be 0 or \pi as shown in Figure (part a) below. The various Fourier theorems provide a thinking vocabulary'' for understanding elements of spectral analysis. Assume that we denote the data sequence x(nT) as x[n] . \cos(A+B) \right\}, the Q component of its DFT is given by DFT size, N Complex multiplications Digital Signal Processing 23 Lecture 2 FFT (RADIX-2) OBSERVATION • Length Nsequence x(n), X(k)=FFTN[x(n)] – even elements: xe(m)=x(2m), Xe(k)=FFTN/2[xe(m)] – odd elements: xo(m)=x(2m+1), Xo(k)=FFTN/2[xo(m)] ⇒ X(k)=Xe(k)+e−j2π k N Xo(k) Digital Signal Processing 24 Lecture 2. This can be checked by plugging in the expression for a complex sinusoid into DFT definition.\end{align}, Let \theta = 2\pi k/N and using the identity \cos(A)\sin(B) = 0.5 \{ \sin(A+B) – \sin(A-B) \}, we get In general, magnitude-phase and IQ plots convey different information, many examples of which we will encounter throughout this text. For example, a radix-2 FFT restricts the number of samples in the sequence to a power of two. 7. If you guessed that these impulses in frequency domain are actually sinc functions sampled at just the peak and zero values, you are right (compute a length-32 DFT of this sequence instead of length-15, for example). For DFT of a rectangular signal, the IQ equations are given in Eq \eqref{eqIntroductionDFTrectangleI} and Eq \eqref{eqIntroductionDFTrectangleQ}, and the magnitude-phase equations in Eq \eqref{eqIntroductionDFTrectangleM} and Eq \eqref{eqIntroductionDFTrectangleP}. DIFFERENCE BETWEEN DITFFT AND DIFFFT. In addition, some FFT algorithms require the input or output to be re-ordered.\end{align*} 9. for an arbitrary number a and orthogonality of \cos(\cdot) and \sin(\cdot) over a complete period. Most of the discussion until now was around the magnitude plots. \begin{align*}\begin{align*} For our example here, a 128-point DFT shows us the detailed content of the input spectrum. However, we have to differentiate $IQ$-plane of time domain from $IQ$-plane of frequency domain. The DFT finds the contributions of analysis frequencies $F = F_S \cdot k/N$ for for $k = -N/2$, $\cdots,-1,0,1,$ $\cdots,N/2-1$ in an input signal, which are all integer multiples of a fundamental frequency $F_S/N$. Figure below illustrates the sampling sequence in time domain for $M=3$ and $N=15$. In our example, an even symmetric signal would have been obtained for $n_s=-(L-1)/2$. The shape of the actual curve is the same sinc function but it is sampled in frequency domain at just the right points — integer frequency bins due to integer number of cycles of the input in time domain. &\cdots &= \sum \limits _{n=n_s} ^{n_s+L-1} -\sin 2\pi\frac{k}{N}n Now the phase plot for $L=7$, $N=16$ and $n_s=-(L-1)/2-1$ has been drawn in Figure below after unwrapping $180^\circ$ phase jumps which arose due to changing $I$ and $Q$ signs. Now referring to its time domain plot, the first sinusoid $s_1(t)$ is positioned around the origin according to its $-90^\circ$ cosine (or $0^\circ$ sine) phase, while the second sinusoid $s_2(t)$ is positioned around the origin according to its $30^\circ$ cosine (or $120^\circ$ sine) phase. \sum \limits _{n=0} ^{N-1} \sin 2\pi \frac{k}{N}n \cdot \cos 2\pi \frac{k}{N}n – \cos 2\pi \frac{k}{N}n \cdot \sin 2\pi \frac{k}{N}n Performing a 256-point or 512-point DFT, in our case, would serve little purpose. ]\end{align*}, It can be seen that $s[n]$ has only an $I$ component with zero $Q$ component. When an input signal contains a complex sinusoid of peak amplitude $A$ with an integral number of cycles over $N$ input samples, the output magnitude of the DFT for that particular sinusoid is $AN$. \sum \limits _{n=0} ^{N-1} \cos 0 &= N \\ By duality, the inverse is also true: a single impulse in time domain corresponds to an all-ones rectangular sequence in frequency domain. Abstract: An important property of a Zadoff-Chu (ZC) sequence is derived, namely that the discrete Fourier transform (DFT) of a ZC sequence is a time-scaled conjugate of the ZC sequence, multiplied by a constant factor. If the input has a signal component at some intermediate frequency between these integer multiples of $F_S/N$, say $1.3 F_S /N$, the orthogonality does not hold and this input signal shows up to some degree in all $N$ output bins of our DFT due to unaligned sampling instants in frequency domain. Linearity. For the above example, s[0] = x[0]h[0]+x[1]h[3]+x[2]h[2]+x[3]h[1] = 210+040+330+( 1)20 = 90: Shift the folded sequence by rotating CCW around the … The N-point DFT is equal to . Although a mathematical trick (known as L’Hôpital’s rule) can be applied to compute it mathematically, we take an easier course. The DFT $S[k]$ can be found by applying the DFT definition and is plotted in Figure below.\label{eqIntroductionRectangular}\end{align*}. The DFT of a rectangular signal has a mainlobe centered about the $k = 0$ point. Plugging in the expression back for $\theta$, In our example, these peaks turn out to be $0$ and $\pm 5$ as shown in Figure above. We are assuming $N/M$ as an integer here because otherwise $p[n]$ will not be periodic. These $\cos(\cdot)$ and $\sin(\cdot)$ are orthogonal to each other over a complete period, and $\cos(\cdot)$ ($\sin(\cdot)$) of one frequency $k/N$ is also orthogonal to $\cos(\cdot)$ ($\sin(\cdot)$) of other frequencies $k’/N$, provided that $k’$ are also integers, i.e., The DFT of a general sinusoid can be derived similarly by plugging the expression of a complex sinusoid in DFT definition and following the same procedure as in the rectangular sequence example.\end{align}. It is actually made up of all $N$ sinusoids just like a periodic square wave. This is called windowingand a rectangular sequence is the simplest form of a window. This is called windowing and a rectangular sequence is the simplest form of a window. k_{\text{peak}} &= \text{integer} \cdot \frac{N}{M} the actual spectrum has been scaled down by a factor of $1/M=1/3$, __|_____|_____|_____|_____|_____|_____|_____|__. \cos 2\pi \frac{k}{N}n &= \cos 2\pi \frac{0}{N}n = 1, \quad\quad n = 0,\cdots,N-1\\ The corresponding figures are drawn in the Figure above with single impulses at bin 0. One of the reasons is that any signal with a finite duration, say TT seconds, in time domain (that all practical signals have) can be considered as a product between a possibly infinite signal and a rectangular sequence with duration TT seconds. Applying the definitions of magnitude and phase to Eq \eqref{eqIntroductionDFTrectangleI} and Eq \eqref{eqIntroductionDFTrectangleQ} and using $\cos^2 A + \sin^2 A = 1$, we get the magnitude and phase of the DFT of a rectangular signal. \sum \limits _{n=0} ^{N-1} \cos 2\pi \frac{k}{N}n \cdot \sin 2\pi \frac{k}{N}n = 0 \nonumber\\ Theorem 1 Defining the DTFT of a sampled sequence in terms of the FT of the sampled waveform. The question is: How can such complex equations generate such simple figures? Execution time for fft depends on the length, n, of the DFT it performs; see the fft … Observe from Eq \eqref{eqIntroductionDFTSamplingSeqM} that when the denominator is non-zero, the spectrum is clearly zero. For a real sinusoid, $s_Q[n]$ is $0$ and only the first term in $I$ part of the above equation survives. The shape of the sinc function did not change as compared to a $4$ kHz input but it is sampled in frequency domain at non-ideal points — fractional frequency bins due to non-integer number of cycles of the input in time domain. In this case, fft pads the input sequence with zeros if it is shorter than n, or truncates the sequence if it is longer than n. If n is not specified, it defaults to the length of the input sequence. \begin{align*} \sum \limits _{n=0} ^{N-1} \cos 2\pi \frac{k}{N}n \cdot \cos 2\pi \frac{k’}{N}n = 0 \label{eqIntroductionOrthogonality}\\ This result has many practical applications. Starting from $k = 0$, the phase shift can be seen to be $0$. Let us compute the $N$-point DFT of a length-$L$ rectangular sequence\end{equation*}, Your email address will not be published. X(ejω)=11−14e−jω=11−0.25cos⁡ω+j0.25sin⁡ω ⟺X∗(ejω)=11−0.25cos⁡ω−j0.25sin⁡ω Calculating, X(ejω).X∗(ejω) =1(1−0.25cos⁡ω)2+(0.25sin⁡ω)2=11.0625−0.5cos⁡ω 12π∫−ππ11.0625−0.5cos⁡ωdω 12π∫−ππ11.0625−0.5cos⁡ωdω=16/15 We can see that, LHS = RHS.HenceProved [Rectangular signal] From a general rectangular sequence, it is evident that a unit impulse is also a rectangular sequence with length $L = 1$. Signal Processing for Software Defined Radio (SDR).\end{align} The sampling sequence is a sequence of unit impulses repeating with a period $M$ within our observation window (and owing to DFT input periodicity, outside the observation window as well). &= \frac{1}{2\sin \theta/2} \left[ \sin \left( n_s+L-\frac{1}{2}\right)\theta – \sin \left( n_s-\frac{1}{2}\right)\theta \right] \\ k = 0 &\Rightarrow 8000 \cdot\frac{0}{8} = 0~ \text{kHz} \\ Its amplitude cannot be found by plugging in $k=0$ in Eq \eqref{eqIntroductionDFTrectangleM} because both the numerator and denominator become zero resulting in undefined $0/0$ expression. Time reversal of a sequence . It means that the sequence is circularly folded its DFT is also circularly folded. It requires NxN complex multiplications and N(N+1) complex … Windowing is a key concept in implementation of many DSP applications.\end{align*}, In this case, $N$ was equal to $16$.&= \sum \limits _{n=n_s} ^{n_s+L-1} \cos 2\pi\frac{k}{N}n \label{eqIntroductionDerivation1} \\ the effect of time shift in frequency domain, Why Building an SDR Requires DSP Expertise, A Real-Imaginative Guide to Complex Numbers, Discrete Fourier Transform (DFT) as a Filter Bank, Phase Locked Loop (PLL) in a Software Defined Radio (SDR). Assume a signal with frequency $4$ kHz at a sample rate of $F_S = 16$ kHz. Since the sequence x(n) is splitted N/2 point samples, thus. s[n] &= \sin(2\pi 1000 nT_S) + 0.75\sin(2\pi 2000 nT_S + 120^\circ) \nonumber \\ Since $p[n]=0$ except when $n=M$, we can write $p[Mm] = 1$ where $m=0,1,\cdots,N/M-1$. Having known the DFT of a rectangular signal, we have two ways to find the Fourier transform of a unit impulse. Quite a few people use WN for W.\end{align*}. Do remember that the plots are — once again — sinc functions overlapping each other. To comprehend the reason why energy appeared in other frequency bins as well, consider that as explained in the article about complex numbers, a complex sinusoid is composed of $\cos(\cdot)$ and $\sin(\cdot)$ as $I$ and $Q$ components, respectively.\begin{align}  Thus, the output magnitude of the DFT is proved as $AN$. In this video, it demonstrates how to compute the Discrete Fourier Transform (DFT) for the given Discrete time sequence x(n)={0,1,2,3} The discrete Fourier transform (DFT) is a method for converting a sequence of N N N complex numbers x 0, x 1, …, x N − 1 x_0,x_1,\ldots,x_{N-1} x 0 , x 1 , …, x N − 1 to a new sequence of N N N complex numbers, X k = ∑ n = 0 N − 1 x n e − 2 π i k n / N, X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i … A rectangular sequence, both in time and frequency domains, is by far the most important signal encountered in digital signal processing. Now let us change the input signal slightly by making the input frequency equal to $3.7$ kHz as shown in Figure below. In most situations, magnitude-phase plot delivers a great deal of information while in some others, $IQ$ plot is more relevant. The Time reversal property states that if. a $0$ phase at a particular frequency $F = F_S \cdot k/N$ is with respect to a cosine wave at that same frequency. The language of signal processing is simpler.\begin{align} Using the identities $\cos A \cos B$ $+$ $\sin A \sin B$ $=$ $\cos (A-B)$ and $\sin A$ $\cos B$ $-$ $\cos A \sin B$ $=$ $\sin (A-B)$. &= \frac{\sin L \theta/2}{\sin \theta/2} \cos \left(n_s + \frac{L-1}{2}\right)\theta \nonumber S_Q[k] &= \sum \limits _{n=n_s} ^{n_s+L-1} -s_I[n] \sin 2\pi\frac{k}{N}n \\ Circular Time shift . \pi L \frac{k_{zc}}{N} &= \pi \\ A zero crossing right at sample $1$ illustrates that it was sampled at peak value for bin $0$ and at zero for all other bins. \Delta \theta(\pm1) &= 2\pi \frac{\pm1}{N} \times \frac{180^\circ}{\pi} = \pm 22.5^\circ \\

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