496 Practical MATLAB® Applications for Engineers
>> % evaluation of DFT using equations>> F0 =1+2*exp(-j*4*pi*0/5)+4*exp(-j*6*pi*0/5)+2*exp(-j*8*pi*0/5)
F0 =
9>> F1=1+2*exp(-j*4*pi/5)+4*exp(-j*6*pi/5)+2*exp(-j*8*pi/5)
F1 =
-3.2361 + 3.0777i>> F2 =1+2*exp(-j*4*pi*2/5)+4*exp(-j*6*pi*2/5)+2*exp(-j*8*pi*2/5)
F2 =
1.2361 - 0.7265i>> F3 =1+2*exp(-j*4*pi*3/5)+4*exp(-j*6*pi*3/5)+2*exp(-j*8*pi*3/5)F3 =
1.2361 + 0.7265i>> F4 =1+2*exp(-j*4*pi*4/5)+4*exp(-j*6*pi*4/5)+2*exp(-j*8*pi*4/5)F4 =
-3.2361 - 3.0777iNote that the DFT coeffi cients obtained by using the fft command are identical
to the analytical results using the equations of R.5.110.
R.5.112 Recall that the linear convolution of a fi nite sequence h(n) of length N with an
arbitrary sequence f(n) can be evaluated by using two algorithms known as
a. Overlap-add method
b. Overlap-save method
R.5.113 The overlap-add method is based on segmenting the sequence f(n) into a fi nite
number of continuous segments each of length M.
The output sequence g(n) is obtained by convoluting the impulse response h(n)
of length N with each of the segments of f(n), and the partial results are added, as
follows:
g(n)f(n)⊗h(n)
and since
f(n)f (m) 12 f (m)
fi m
i 1∑ ()
gn f mi hn
i() ( ) ()
1∑
⊗
gn f mi hn
i() [ ( ) ()]
1∑ ⊗