A. Data Types and Data Structures - an introduction to the next 5 lessons

• Computers store and manipulate data. Since there are different types of data in the world (e.g., text, whole numbers, fractions, music, video, etc.), computer programming languages implement a variety of data types to correctly deal with these different types of data.
   
• A data type defines 3 major properties of a data value:
  • The amount of memory (number of bits) used for each unique value of this data type.
  • What the memory bits mean -- which determines the range of possible values the data type can represent.
  • What valid operations are permitted on values of this data type.
     
• The basic data types in MATLAB are:
   

  Type of Data	Data Type	Number of bits (bytes) used for one value	Range of values	Valid operations
  real numbers	double	64 (8)	()2.2251 x 10-308  to      1.7977 x 10308	arithmetic and vector algebra operations
  real numbers	single	32 (4)	()1.17549 x 10-38 to      3.40282 x 10+38	arithmetic and vector algebra operations
  logical (true/false) values	logical	1 (1)	0 or 1	logical operators
  whole numbers	int8	8 (1)	-128 to +127	arithmetic operations
  whole numbers	int16	16 (2)	-32,768 to + 32,767	arithmetic operations
  whole numbers	int32	32 (4)	-2,147,483,648 to +2,147,483,647	arithmetic operations
  positive whole numbers	uint8	8 (1)	0 to 255	arithmetic operations
  positive whole numbers	uint16	16 (2)	0 to 65,535	arithmetic operations
  positive whole numbers	uint32	32 (4)	0 to 4,294,967,295	arithmetic operations
  text	char	16 (2)	Unicode character encodings	arithmetic operations and string functions

   

• A data structure is a method of organizing multiple values into a single structure that facilitates data manipulate. It is not uncommon that data organized in one way is difficult to process, but organized in another way it becomes easier to process. It is a programmer's job to select appropriate data structures for their data based on the problem being solved.
   
• The basic data structures in MATLAB are:
   

  Data Structure	MATLAB name	Comments
  scalar	1-by-1 array	single values
  row vector	1-by-n array	list of values
  column vector	n-by-1 array	list of values
  matrix	n-by-m array	grid of values
  multidimensional array	n-by-m-by-p array n-by-m-by-p-by-s array etc.	more than 2 indexes required to select a unique value out of the array
  sparse array	(any of the above)	an efficient way to store large arrays that contain very few non-zero elements
  cell array	(any of the above)	a collection of values, each possibly of a different data type and data structure
  structure array	a list of "named" values	a collection of values, each possibly of a different data type and data structure, that are identified by a unique field "name."

   

B. Complex numbers

• Review of complex numbers

• Complex numbers come from the fact that a simple equation like r² + 1 = 0 has no solution from the real numbers. If you solve for r, the "answer" is the square root of -1, which does not exist. So mathematicians created a value for the square root of -1 and called it "i". But the square root of -1 does not exist! (You can't have everything.) Using i allows us to calculate answers to many equations that would not have any solutions otherwise. (Isn't it great when you can just make up things to get answers!).
• Complex numbers are critical to the analysis of many types of engineering systems, especially those that include phase shifts like signal analysis.
   
• (a + bi) is a complex number, where a and b are real values and i is the square root of -1.
• If plotted in rectangular coordinates, the value a is associated with the x-axis and the value b is associated with the y-axis.
• The magnitude of a complex number (a + bi) is sqrt(a² + b²).
• The magnitude of a complex number is equal to the distance from the origin of the (x,y) point (a,b).
• The angle of (a + bi) is tan^-1(b/a).

• Standard operations on complex numbers:    assume that c₁ = (a₁ + b₁i) and c₂ = (a₂ + b₂i)

• c₁ + c₂ = (a₁ + a₂) + (b₁ + b₂)i
• c₁ - c₂ = (a₁ - a₂) + (b₁ - b₂)i
• c₁ * c₂ = (a₁a₂ - b₁b₂) + (a₁b₂ + b₁a₂)i
• c₁ / c₂ = (a₁a₂ - b₁b₂)/(a₂² + b₂²) + (b₁a₂ - a₁b₂)/(a₂² + b₂²)i
  
  for example, execute these statements in MATLAB and examine the results
      c = 2 + 3i
      d = 3 + 4i
      disp(c + d)
      disp(c - d)
      disp(c * d)
      disp(c / d)

• MATLAB functions (a subset)  that return information about complex numbers

• real(c)     returns the real part of a complex number c
• imag(c)     returns the imaginary part of a complex number c
• isreal(c)   returns true (1) if c is not complex
• abs(c)      returns the magnitude of a complex number
• angle(c)    returns the angle of a complex number

• Plotting complex numbers

• given the equations

t = 0:pi/20:4*pi;
y = exp(-0.2*t) .* (cos(t) + i*sin(t)); % any function that creates complex values
 

plot the real part only	plot(t, y)
plot the imaginary part only	plot(t, imag(y))
plot real vs. imaginary	plot(y)
plot the complex values in polar coordinates	polar(angle(y), abs(y))

C. Integer data types

• Integer values represent whole numbers with no fractional part.
   
• In MATLAB, integer values are used primary to represent image and sound data.
   
• Integer values use much less storage (memory) than double precision floating point values. If an application needs to manipulate large data sets, storing the data in an integer format can reduce memory usage.
   
• The types of integer values are shown below (Note that 'u' stands for 'unsigned', which means that all values are positive.)

Data type	Range of possible values	Memory usage per value
int8	-128 to +127	1 byte
uint8	0 to 255	1 byte
int16	-32,768 to +32,767	2 bytes
uint16	0 to 65,535	2 bytes
int32	-2,147,483,648 to +2,147,483,647	4 bytes
uint32	0 to 4,294,967,295	4 bytes

• By default all values in MATLAB are stored using the double data type. To change a value to a different data type, use a cast operation. All casts operators have the same name as their associated data type. For example:
   
  a = int8(5) % a is now an int8 variable
  whos a

  b = 2.8;
  a = int16( (5.2 + b) / 2.5 ) % does double precision arithmetic and then converts the results to a 16 bit integer
  whos a
   

• If all of the variables in an expression are of the same data type, then the results of the expression will be a value of that same data type. Execute the following two examples and examine the results:
   
  a = int8(5)
  b = int8(6)
  c = a + b
  whos a b c

  a = uint16(3280)
  b = uint16(4376)
  c = a + b
  whos a b c
   

• A mixed mode expression is any expression that contains variables and values of different data types.
  • Only one integer data type can be used in a single expression.
  • A mixed mode expression can contain only one integer data type and doubles.
  • The result of a mixed mode expression will always result in an integer value of the same integer data type used in the expression.
  • Any double precision values in a mixed mode expression are rounded to their closest integer value before any operations are performed.
  • Division using integer values rounds the results to the closest integer.
     
  • Execute the following examples and examine the results:
     
  a = int8(5) + int16(5);  % generates a error because it contains two different integer data types

  a = int8(5)
  b = 6.7 
  c = a + b  % results in 12 because the the 6.7 is rounded to 7 before the addition

  a = int8(10) / int8(15) % results in 1 because the division result is rounded to the closest integer
   
• If integer values are manipulated, and the results become too large (or too small) for the data type being used, the results are clamped to the largest (or smallest) possible value for that data type. For example:

a = int8(100);
b = int8(100);
disp(a + b)

c = uint16(10000);
d = uint16(10);
disp(c * d)

• Most MATLAB built-in functions work on integer data types, but they typically return a value that is not an integer. Some MATLAB functions will not process integer data at all. For example:
   
  a = sum( int8( [1 2 3 4 5 6] ) ) % returns an answer of type "double"

  a = mean( int8( [1 2 3 4 5 6] ) ) % returns an answer of type "double"

  a = sqrt( int8( [1 2 3 4 5 6] ) ) % generates an error message - invalid operation for integer data