A.18  Fourier Analysis: Pairs

This section lists few pairs, which are among the most important ones. Both continuous and discrete-time signals are exemplified.

f 1.

impulse $\iff$ DC level

$$x(t)=\delta (t)\iff X(\omega )=2\pi$$

$$x[n]=\delta [n]\iff X(e^{j\mathrm{\Omega }})=1$$

$$x[n]=\frac{1}{2\pi }\iff X(e^{j\mathrm{\Omega }})=\text{∑}_{k=-\infty }^{\infty }\delta (\mathrm{\Omega }-k2\pi )$$

f 2.

pulse $\iff$ sinc

Several pairs of pulses and sincs are described below.⁴

$$x(t)=\{\begin{array}{c}\hfill A,-T∕2\le t\le T∕2\hfill \\ \hfill 0,\mathrm{\text{otherwise}}\hfill \end{array}\iff X(f)=\mathit{AT}\mathrm{\text{sinc}}(\mathit{fT})$$

(A.54)

 

$$x[n]=\{\begin{array}{c}1,0\le n\le M-1\hfill \\ 0,\mathrm{\text{otherwise}}\hfill \end{array}\iff X(e^{j\mathrm{\Omega }})=\frac{\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(\mathrm{\Omega }M∕2)}{\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(\mathrm{\Omega }∕2)}{e}^{-j\mathrm{\Omega }(M-1)∕2}$$

(A.55)

Due to the duality property, sincs in time-domain lead to pulses in frequency-domain. In continous-time, one has:

$$x(t)=\frac{1}{\mathit{\pi t}}\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(2\mathit{\pi Ft})=2F\mathrm{\text{sinc}}(2\mathit{Ft})\iff X(f)=\{\begin{array}{cc}1,\hfill & \hfill |f|\le F\\ 0,\hfill & \hfill \mathrm{\text{otherwise}}\end{array}$$

(A.56)

where $F$ is given in Hertz, and

$$x(t)=\frac{1}{\mathit{\pi t}}\mathrm{\text{sin}}(\mathit{Wt})\iff X(\omega )=\{\begin{array}{cc}1,\hfill & \hfill |\omega |\le W\\ 0,\hfill & \hfill \mathrm{\text{otherwise}}\end{array}$$

(A.57)

where $W$ is given in rad/s. Note that in Eq. (A.57) one has a sine, not a sinc.

Considering discrete-time signals and assuming $\alpha \ge 1$ determines the spectrum bandwidth (recall that it suffices to specify $X(e^{j\mathrm{\Omega }})$ for $-\pi \le \mathrm{\Omega }\le \pi$ ) one has:

$$x[n]=\frac{2\pi }{\alpha }\mathrm{\text{sinc}}\left(\frac{n}{\alpha }\right)\iff X(e^{j\mathrm{\Omega }})=\{\begin{array}{c}1,-\pi ∕\alpha \le \mathrm{\Omega }\le \pi ∕\alpha \hfill \\ 0,\mathrm{\text{otherwise}}\hfill \end{array}$$

(A.58)

Now it is assumed a discrete-time pulse train $x[n]$ with period $N$ and, for the pulse centered in 0, $x[n]=1$ from $n=-M$ to $N=M$ and 0 otherwise. This pulse is replicated: the next is centered in $n=N$ and has non-zero values in the range $[N-M,N+M]$ and so on. The spectrum is

$$X[k]=\frac{1}{N}\frac{\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\left(k(2M+1)\frac{\pi }{N}\right)}{\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\left(k\frac{\pi }{N}\right)},$$

where $X[k]=\frac{2M+1}{N},k=0,\pm N,\pm 2N,\dots$, via L’Hopital’s rule.

For a continuous-time pulse train with each pulse of duration $2T_p$ and period $T_0$ (one pulse is centered at $t=0$, with duration from $-T_p$ to $T_p$) one has:

$$c_k=\frac{\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(2\mathit{\pi k}{f}_0{T}_p)}{\mathit{k\pi }}=\frac{\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(k{\omega }_0{T}_p)}{\mathit{k\pi }}=2f_0{T}_p\mathrm{\text{sinc}}(2k{f}_0{T}_p)=\frac{2T_p}{T_0}\mathrm{\text{sinc}}\left(\frac{k2T_p}{T_0}\right),$$

where $f_0=1∕T_0$ and ${\omega }_0=2\pi f_0$.

f 3.

exponential $\iff$ rational function

$$e^{-\mathit{at}}u(t)\iff \frac{1}{(\mathit{j\omega }+a)},a>0$$

 

$$a^{n}u[n]\iff \frac{1}{(1-a{e}^{-j\mathrm{\Omega }})},\left|a\right|<1$$

f 4.

complex exponential $\iff$ impulse

$$e^{j2\pi f_{0}t}\iff \delta (f-f_0)$$

(A.59)

f 5.

train of impulses $\iff$ train of impulses

The Fourier series coefficient of an impulse train of period $T_0$, with one of the impulses $\delta (t)$ at the origin $t=0$ is $c_k=1∕T_0$. In case the impulses are shifted in time, a linear phase appears in $c_k$. If the series should be represented in the transform domain, the coefficient values $c_k$ are represented by impulses with area $c_k$ and separated in frequency by multiples of $f_0=1∕T_0$, creating another impulse train:

$$\sum _{l=-\infty }^{\infty }\delta (t-l{T}_0)\iff \frac{1}{T_0}\sum _{k=-\infty }^{\infty }\delta \left(f-\frac{k}{T_0}\right).$$

(A.60)

In rad/s instead of Hertz, the pair is:

$$\sum _{l=-\infty }^{\infty }\delta (t-l{T}_0)\iff \frac{2\pi }{T_0}\sum _{k=-\infty }^{\infty }\delta \left(\omega -\frac{k2\pi }{T_0}\right)$$

(A.61)

For discrete-time, a train of impulses with amplitude one and period $N_0$ leads to series coefficients $X[k]=1∕N_0,\forall <!--\; FUNCTION\; APPLICATION\; -->k$. Representing these coefficients in the transform domain leads to another train of impulses in $\mathrm{\Omega }$ spaced by $2\pi ∕N_0$ with areas $2\pi ∕N_0$:

$$\sum _{l=-\infty }^{\infty }\delta [n-l{N}_0]\iff \frac{2\pi }{N_0}\sum _{k=-\infty }^{\infty }\delta \left(\mathrm{\Omega }-k\frac{2\pi }{N_0}\right).$$

Due to the corresponding Fourier series, note that any periodic impulse train can be written as a sum of of complex exponentials. For instance, in discrete-time:

$$\sum _{l=-\infty }^{\infty }\delta [n-l{N}_0]=\frac{1}{N_0}\sum _{k=0}^{N_0-1}{e}^{j2\mathit{\pi kn}∕N_0}.$$

(A.62)

In continuous-time, one can represent the transform $X(f)$ of the impulse train in two distinct ways. One is by calculating the series coefficients $c_k=1∕T_0$ and then representing them in the transform domain. The second alternative is via the definition of the Fourier transform of the impulse train, and then using the impulse sifting property. These two alternatives can be written as:

$$X(f)=\frac{1}{T_0}\sum _{k=-\infty }^{\infty }\delta \left(f-\frac{k}{T_0}\right)=\sum _{k=-\infty }^{\infty }{e}^{-j2\mathit{\pi fk}{T}_0}.$$

(A.63)

The equality in Eq. (A.63) is not trivial because the continuous-time impulse is a generalized function. In order to get insight on how the sum of these complex exponentials converge to an impulse train, the interested reader can use the code MatlabOctaveCodeSnippets/snip_appfourier_impulse_train.m.