Discrete Signal Processing (DSP):

Discrete Signals


By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH

Table of Contents

1. Discrete Time Signals¶

A signal $x[n]$ is a function that maps an independent variable to a dependent variable.

In this course, we will focus on discrete-time signals $x[n]$:

  • Independent variable is an integer: $n \in \mathbb{Z}$
  • Dependent variable is a real or complex number: $x[n] \in \mathbb{R}$ or $\mathbb{C}$




1.1. Plot Real Signals¶

  • plot for continuous signals in MATLAB

  • stem for discrete signals in MATLAB

$$x(t) = \sin(2\pi t)$$
In [2]:
%plot -s 560,200

t = 0:0.01:2;
x = sin(2*pi*t);

plot(t, x, 'linewidth', 2);
ylim([-1.1 1.1]);
xlabel('t [sec]');
ylabel('x(t)');










    



























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











In [3]:



    


%plot -s 560,400

N = 20;
n = 0:N-1;
x = sin(2*pi/N*n);

subplot(2,1,1)
plot(n,x,'o'), axis tight, ylim([-1.2, 1.2])
subplot(2,1,2)
stem(n,x,'filled'), axis tight, ylim([-1.2, 1.2])



    















    















    




























2. Signal Properties¶
2.1. Finite/Infinite-Length Signals¶
    

  • An infinite-length discrete-time signal $x[n]$ is defined for all $n \in \mathbb{Z}$, i.e., $-\infty < n < \infty$
    • 










    

  • An finite-length discrete-time signal $x[n]$ is defined only for a finite range of $N_1 < n < N_2$
    • 











2.2. Periodic Signals¶
    

  • A discrete-time signal is periodic if it repeats with period $N \in \mathbb{Z}$
    • 





$$x[n+mN] =  x[n], \qquad \forall m \in \mathbb{Z}$$










    

  • The period $N$ must be an integer
    • 



    

  • A periodic signal is infinite in length
    • 



    

  • Convert an infinite-length signal $x[n]$ into a finite-length signal $y[n]$ by windowing
    • 





$$y[n] = \begin{cases}x[n] & N_1 \leq n \leq N_2\\ 0 & \text{otherwise} \end{cases}$$



    

  • Convert a finite-length signal $x[n]$ defined for $N_1 \leq n \leq N_2$ into an infinite-length signal by either
      

    • (infinite) zero padding, or

      $$y[n] = \begin{cases} 0 & n < N_1\\ x[n] &N_1 \leq n \leq N_2\\ 0 & N_2 < n\end{cases}$$
      • 

    • periodization with period $N$

      $$\begin{align*}y[n] &= \sum_{m=-\infty}^{\infty}x[n-mN]\\&=\cdots + x[n+N] + x[n] + x[n-N]+ \cdots\end{align*}$$
      • 

    

    • 
























2.3. Modular Arithmetic¶
    

  • Modular arithmetic with modulus $N$ takes place on a clock with $N$
      

    • Modular arithmetic is inherently periodic
$$\cdots = (-12)_8 = (-4)_8 = (4)_8 = (12)_8 = (20)_8 = \cdots$$
      • 

    

    • 











    

  • Periodization via Modular Arithmetic
      

    • Consider a length-$N$ signal $x[n]$ defined for $0 \leq n \leq N-1$
      • 

    • A convenient way to express periodization with period $N$ is $y[n] = x[(n)_N]$
      • 

    • Important interpretation
        

      • Infinite-length signals live on the (infinite) number line
        • 

      • Periodic signals live on a circle 
        • 

      

      • 

    

    • 














In [4]:



    


%plot -s 560,100

N = 8;
n = 0:N-1;
x = [0 1 2 3 4 3 2 1];

stem(n,x,'filled');
xlabel('n'), ylabel('x[n]')
xlim([-16,23]), 
yticks([0,4])
box('off')



    















    















    


























In [5]:



    


%plot -s 560,100

%% periodic using mod
y = [];
n = -16:23;

for i = 1:length(n)
    y(i) = x(mod(n(i),N)+1);
end

stem(n,y,'filled'), axis tight
xlabel('n'), ylabel('y[n]')
yticks([0,4])
box('off')



    















    















    




























2.4. Finite-Length and Periodic Signals¶
    

  • Finite-length and periodic signals are equivalent
      

    • All of the information in a periodic signal is contained in one period (of finite length)
      • 

    • Any finite-length signal can be periodized
      • 

    • Conclusion: We will think of finite-length signals and periodic signals interchangeably
      • 

    

    • 











    

  • Shifting infinite-length signals
      

    • Given an infinite-length signal $x[n]$, we can shift back and forth in time via $x[n-m]$
      • 

    • When $m>0, x[n-m]$ shifts to the right (forward in time, delay)
      • 

    • When $m<0, x[n-m]$ shifts to the left (back in time, advance)
      • 

    

    • 











    

  • Shifting periodic signals
      

    • Periodic signals can also be shifted; consider $y[n]=x[(n)_N]$
      • 

    • Shift one sample into the future: $y[n-1] = x[(n-1)_N]$
      • 

    

    • 











    

  • Shifting finite-length signals
      

    • Consider finite-length signals $x$ and $y$ defined for $0 \leq n \leq N-1$ and suppose $y[n] = x[n-1]$
      

      $$
\begin{align*}
y[0] &= \;?\\
y[1] &= x[0]\\
y[2] &= x[1]\\
y[3] &= x[2]\\
&\vdots\\
y[N-1] &= x[N-2]\\
?\; &= x[N-1]\\
\end{align*}
$$
      • 

    • What to put in $y[0]?$ What to do with $x[N-1]$? We do not want to invent/lose information
      • 

    • Elegant solution: Assume $x$ and $y$ are both periodic with period $N$; then $y[n]=x[(n-1)_N]$
      • 

    • This is called a periodic or circular shift
      • 

    

    • 



    

  • Circular time reversal 
     $$y[n]= x[(-n)_N]$$
      

    • Example with $N = 8$
      • 

    

    • 


$$
\begin{align*}
y[0] &= x[0]\\
y[1] &= x[7]\\
y[2] &= x[6]\\
y[3] &= x[5]\\
&\vdots\\
y[7] &= x[1]
\end{align*}
$$