Fourier Analysis of Digital Signals

Abstract

The advances in digital computers after the Second World War led to the appearance of a new engineering branch. Digital signal processing is the name of this engineering branch. Advances in computer technology have led to the design of almost all signal processing methods with digital technology. Analog technology has been almost completely abandoned, and most newly manufactured devices are digitally designed. Signal processing techniques developed for continuous-time signals are adapted to digital signals. Some of these digital signal processing techniques are Fourier series representation of digital signals and digital Fourier transform. The formula for the Fourier series representation of digital signals is obtained using the Fourier series representation formula of continuous-time signals. Despite advances in computer technology, some signal processing algorithms are still time-consuming and need huge computation amount for computers. For this reason, algorithms are modified to run faster by computers. One of the best known of these algorithms is the fast Fourier transform algorithm developed in the 1960s. With the development of this algorithm, there has been an acceleration in the design of systems used for signal processing.

Appendices

Summary

Let x[n] be a periodic signal with period N such that

$$ x\left[n\right]=x\left[n+ mN\right] $$

(5.95)

One period of x[n] denoted by x_op[n] is defined as

$$ {x}_{\textrm{op}}\left[n\right]=\left\{\begin{array}{c}x\left[n\right]\kern1em 0\le n\le N-1\\ {}0\kern2em \textrm{otherwise}\kern1em \end{array}\right. $$

(5.96)

from which we can write that

$$ x\left[n\right]=\sum \limits_{m=-\infty}^{\infty }{x}_{\textrm{op}}\left[n- mN\right] $$

(5.97)

The Fourier series coefficients of x[n] are calculated using

$$ X\left[k\right]=\frac{1}{\sqrt{N}}\sum \limits_{n,N}x\left[n\right]{e}^{- jk{\Omega}_0n}\kern1em {\Omega}_0=\frac{2\pi }{N} $$

(5.98)

which can be used to obtain the Fourier series representation of x[n] as

$$ x\left[n\right]=\frac{1}{\sqrt{N}}\sum \limits_{k,N}X\left[k\right]{e}^{jk{\Omega}_0n} $$

(5.99)

X[k] is a complex periodic digital signal, and it can be considered a vector consisting of complex numbers. Each complex number in X[k] has a magnitude and phase. x_op[n] equals to one period of x[n]. x_op[n] is a non-periodic signal

Fourier transform for digital signals is calculated using

$$ {X}_{\textrm{op}}\left(\Omega \right)=\frac{1}{\sqrt{2\pi }}\sum \limits_{n=-\infty}^{\infty }{x}_{\textrm{op}}\left[n\right]{e}^{-j\Omega n} $$

(5.100)

for which the inverse Fourier transform is calculated using

$$ {x}_{\textrm{op}}\left[n\right]=\frac{1}{\sqrt{2\pi }}{\int}_{2\pi }{X}_{\textrm{op}}\left(\Omega \right){e}^{j\Omega n}d\Omega $$

(5.101)

The Fourier transform function X_op(Ω) obtained for digital signals is a periodic function, and its period equals 2π. The digital signal X[k] can be obtained by sampling X_op(Ω) in the frequency domain. The sampling frequency can be chosen of ${\Omega}_0=\frac{2\pi }{N}$. Thus, we can write that

$$ X\left[k\right]=\sqrt{\frac{2\pi }{N}}{\left.{X}_{\textrm{op}}\left(\Omega \right)\right|}_{\Omega =k{\Omega}_0}\kern1em {\Omega}_0=\frac{2\pi }{N} $$

(5.102)

Problems

• 1. Are the following signals periodic? If they are periodic, find their periods:

  • (a) ${x}_1\left[n\right]=\cos \left(\frac{3\pi }{17}n+\frac{\pi }{3}\right)$

  • (b) ${x}_2\left[n\right]=2\sin \left(\frac{14\pi }{19}n\right)+\cos (n)+1$

  • (c) ${x}_3\left[n\right]={\sum}_{m=-\infty}^{\infty }{\left(-1\right)}^m\delta \left[n-3m\right]$

• 2. $x\left[n\right]=1+{e}^{-j\frac{2\pi }{3}n}\to \left|X\left(\Omega \right)\right|=?$, ∠X(Ω) = ?

• 3. x[n] = δ[n + 1] + 2δ[n − 1] → X(Ω) = ? ,   |X(Ω)| = ? ,  ∠ X(Ω) = ?

• 4. Using x[n] in (2), we obtain the periodic signal $y\left[n\right]=\sum \limits_{l=-\infty}^{\infty }x\left[n-4l\right]$:

  • (a) What is the period of y[n]?

  • (b) Find the Fourier series coefficients of y[n].

• 5. For x[n] = − δ[n + 2] + δ[n + 1] + δ[n − 1] − δ[n − 2], calculate X(Ω) and roughly draw ∣X(Ω)|. If x[n] is accepted as a filter, determine whether x[n] is a low-pass filter or high-pass filter.

• 6. Show that ${x}_1\left[n\right]\ast {x}_2\left[n\right]\overset{\textrm{FT}}{\leftrightarrow}\sqrt{2\pi }{X}_1\left(\Omega \right){X}_2\left(\Omega \right).$

• 7. Let x_op[n] be a non-periodic signal and x[n] be a periodic signal with period N. Prove the following.

  • (a) $\sum \limits_{n=-\infty}^{\infty }{\left|{x}_{\textrm{op}}\left[n\right]\right|}^2={\int}_{2\pi }{\left|{X}_{\textrm{op}}\left(\Omega \right)\right|}^2d\Omega$

  • (b) $\sum \limits_{n=0}^{N-1}{\left|x\left[n\right]\right|}^2={\sum}_{k=0}^{N-1}{\left|X\left[k\right]\right|}^2$

• 8. If $X\left(\Omega \right)=\frac{-\frac{1}{3}{e}^{-j\Omega}+2}{1+\frac{1}{6}{e}^{-j\Omega}-\frac{1}{6}{e}^{-j2\Omega}}$, determine whether x[n] is a periodic signal or not. Find x[n].

• 9. The relationship between input and output of a linear time invariant system is given by

  $$ y\left[n-2\right]-5y\left[n-1\right]+6y\left[n\right]=4x\left[n-1\right]+8x\left[n\right] $$

  where x[n] is the system input and y[n] is the system output:

  • (a) Find the frequency response of the system.

  • (b) Find the impulse response of the system.

• 10. If $x\left[n\right]\overset{\textrm{FT}}{\leftrightarrow }X\left(\Omega \right)$, then find the Fourier transforms of nx[n] and nx[n − n₀] in terms of X(Ω).

• 11. Find the Fourier transforms of the following signals:

  • (a) $x\left[n\right]={\left(\frac{1}{2}\right)}^nu\left[n\right]$

  • (b) $x\left[n\right]=n{\left(\frac{1}{2}\right)}^nu\left[n-3\right]$

  • (c) $x\left[n\right]=n{\left(\frac{1}{2}\right)}^n{e}^{j\frac{\pi }{8}n}u\left[n-3\right]$

  • (d) $x\left[n\right]=\frac{\sin \left(\pi \left(n+1\right)\right)}{\pi \left(n+1\right)}$

  • (e) $x\left[n\right]=\sin \left(\frac{\pi }{2}n\right){\left(\frac{1}{3}\right)}^nu\left[n-3\right]$

  • (f) $x\left[n\right]={\left[\frac{\sin \left(\frac{\pi }{3}n\right)\ }{\pi n}\right]}^3\ast \frac{\sin \left(\frac{\pi }{3}n\right)\ }{\pi n}$

  • (g) $x\left[n\right]=\frac{\sin \left(\frac{\pi }{2}n\right)\ }{\pi n}$

  • (h) $x\left[n\right]=n{\left(\frac{3}{5}\right)}^{\left|n\right|}$

• 12. Find the digital periodic signals whose Fourier series coefficients are given as:

  • (a) $X\left[k\right]=\cos \left(\frac{\pi }{3}k\right)$

  • (b) X[k] = jδ[k − 1] − jδ[k + 1]  Ω₀ = 2π

  • (c) $X\left[k\right]=\cos \left(\frac{4\pi }{21}k\right)$

• 13. Find the inverse Fourier transform of the following signals:

  • (a) X(Ω) = cos (2Ω) + j sin (2Ω)

  • (b) X(Ω) = 2 cos (4Ω)

  • (c) $X\left(\Omega \right)=\sin \left(2\Omega \right)+\cos \left(\frac{\Omega}{2}\right)$

  • (d) $X\left(\Omega \right)=\frac{3}{-3-{2}^{-j\Omega}+{e}^{-j2\Omega}}$

• 14. The relationship between input and output of a linear time invariant system is given by

  $$ y\left[n-2\right]+7y\left[n-1\right]+6y\left[n\right]=8x\left[n-1\right]+10x\left[n-2\right] $$

  Find the impulse response of this system.

• 15. $x\left[n\right]={\left(-1\right)}^n,\kern0.5em h\left[n\right]=\frac{\sin \left(\frac{\pi }{3}n\right)}{\pi n}\to y\left[n\right]=x\left[n\right]\ast h\left[n\right],\kern0.5em y\left[n\right]=?$

• 16. $x\left[n\right]={\left(-1\right)}^{n+1},\kern1em h\left[n\right]=\delta \left[n\right]-\frac{\sin \left(\frac{\pi }{3}n\right)}{\pi n}\to y\left[n\right]=x\left[n\right]\ast h\left[n\right],\kern0.5em y\left[n\right]=?$

• 17. The relationships between the Fourier transforms of x[n], i.e., X(Ω), and y[n], i.e., Y(Ω) are given as:

  • (a) Y(Ω) = e^−j2ΩX(Ω)

  • (b) Y(Ω) = Re {X(Ω)}

  • (c) $Y\left(\Omega \right)=\frac{d^2X\left(\Omega \right)}{d{\Omega}^{2.}}$

  Express y[n] in terms of x[n] for each case.

• 18. Find the Fourier transform of $x\left[n\right]=\sin \left(\frac{\pi }{2}n\right)+\sin \left(\frac{\pi }{3}n\right)$.

• 19. For periodic signal x[n], show that if

  $$ x\left[n\right]\overset{\textrm{DTFSC}}{\leftrightarrow }X\left[k\right] $$

  then we have

$$ X\left[n\right]\overset{\textrm{DTFSC}}{\leftrightarrow }x\left[-k\right] $$