Introduction

A real, N-periodic, discrete-time signal  can be represented by a linear combination of the complex exponential signals

 

 

as

 

In these expressions, , and the discrete-time fundamental frequency is . This discrete-time Fourier series representation provides notions of frequency content of periodic, discrete-time signals, and it is very convenient for calculations involving linear, time-invariant systems because complex exponentials are eigenfunctions of LTI systems.

 

The complex coefficients  can be calculated from the expression

 

 

The  are called the spectral coefficients of the signal . A plot of  vs k is called the magnitude spectrum of , and a plot of  vs k is called the phase spectrum of . These plots, particularly the magnitude spectrum, provide a picture of the frequency composition of the signal. Notice that the spectral coefficients repeat as k is varied. In particular, for any value of k,

 

 

First Applet - Entering Signals

First Applet - Entering Coefficients or Spectra

 

This applet illustrates the discrete-time Fourier series representation for N = 5, that is, . In addition to a “live” mathematical expression for the signal, display windows show

·        two repetitions of the magnitude and phase spectra,

·        the individual frequency components (often called phasors) in the complex plane,

·        the sum of these phasor components in the complex plane,

·        two periods of the signal .

 

First, select one of three ways to enter data. You can enter values for the magnitudes (in the range ) and angles (in the range  radians) of coefficients. To update the resulting expression, click outside the text field. Alternatively, you can enter the magnitude and phase spectra by sketching with the mouse, or you can enter a signal  with the mouse. Then select play to observe the frequency components and the generation of the signal from these components.

 

For , the general Fourier series expression can be written as

 

 

Exercises

 

(1) Repetition of the spectral coefficients as shown above implies that . What other patterns or symmetries do you observe in the magnitude spectrum? Justify your answer mathematically.

 

(2) Repetition of the spectral coefficients implies that . What other patterns or symmetries do you observe in the phase spectrum? Justify your answer mathematically.

 

(3) Using (1) and (2), explain why  and  for every (real) signal .

 

(4) Suppose  is even, that is, . What can you conclude about the spectral coefficients? Can you justify your answer mathematically? (For convenience of signal entry, use the periodicity property, , to express the even property as . )

 

 (5) Suppose has exactly one nonzero value per period. What do you observe about the magnitude spectrum? Does it matter where the nonzero value occurs? Justify your answer mathematically.

 

(6) If  has exactly one nonzero value per period, what do you observe about the phase spectrum? Does it matter where the nonzero value occurs?

 

Second Applet

 

For this applet, you can enter a value of N between 4 and 32, and then enter either a signal or the frequency spectra. Only one period of the signal and one repetition of the spectra are shown.

 

 

Exercises

 

(1) If the period N is an even integer and , what pattern do you observe in the magnitude spectrum? Justify your answer mathematically.

 

(2) If the N is an even integer and, what pattern do you observe in the magnitude spectrum? Justify your answer mathematically.

 

(3) If N = 20, what frequencies correspond to the spectral coefficients  for k = 0, 9, and 19? Which of these frequencies would you call “high” frequencies and which would you call “low?”

 

(4) If N = 20, what is the signal that has all spectral coefficients zero except ? What is the signal if ?

 

(5) If N is divisible by 4, what are the spectral coefficients corresponding to sine waves with periods of N, N/2, N/4 ?

 

 

return to demonstrations page