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14) (MATLAB) In some cases, to calculate the convolution of two sequences, the convolution theorem of Discrete Fourier Transformation may be preferable instead of the direct application of the convolution summation. This is due to the smaller number of arithmetic operations (i.e. additions and multiplications) of the DFT based method compared to the number of arithmetic operations of the convolution summation. The small number of arithmetic operations of the DFT is achieved by using an efficient implementation of the DFT, namely FFT algorithm, which will be discussed at the end of the term. i) Generate a $N_1$-point sequence, $x_1[n]$ and an $N_2$-point filter $h[n]$, by using round(randn(1,N)) command in MATLAB. Select $N_1$=8192 first and then you may try larger values such as $N_1=2^{15}$ etc. Select $N_2$=100. ii) Obtain the convolution result $y_1[n]$=conv($x_1$,h). iii) Repeat ii) by using the convolution theorem of DFT (note you can calculate dft using fft(x,L) where L is DFT length you desire) to obtain $y_2[n]$ as the result of the convolution of these two sequences. (Note that you need to select an appropriate length for the DFT to obtain linear convolution instead of circular convolution). iv) Verify that $y_1[n]=y_2[n]$. v) What is the DFT length you used in iii)? In some applications such large DFT sizes may not be possible due to computational constraints or real-time processing requirements (i.e. you may not be allowed to wait to store such long sequences before processing). Hence, overlap-add or overlap-save methods can be used instead. Implement the overlap-add method where you are allowed to use DFT and IDFT of only length 256. What is the segment size range you can use? Let the result with overlap-add method be $y_3[n]$ and verify it equals $y_1[n]$ or $y_2[n]$.

          14) (MATLAB) In some cases, to calculate the convolution of two sequences, the convolution theorem
of Discrete Fourier Transformation may be preferable instead of the direct application of the
convolution summation. This is due to the smaller number of arithmetic operations (i.e. additions and
multiplications) of the DFT based method compared to the number of arithmetic operations of the
convolution summation. The small number of arithmetic operations of the DFT is achieved by using an
efficient implementation of the DFT, namely FFT algorithm, which will be discussed at the end of the
term.
i) Generate a $N_1$-point sequence, $x_1[n]$ and an $N_2$-point filter $h[n]$, by using round(randn(1,N))
command in MATLAB. Select $N_1$=8192 first and then you may try larger values such as $N_1=2^{15}$
etc. Select $N_2$=100.
ii) Obtain the convolution result $y_1[n]$=conv($x_1$,h).
iii) Repeat ii) by using the convolution theorem of DFT (note you can calculate dft using fft(x,L)
where L is DFT length you desire) to obtain $y_2[n]$ as the result of the convolution of these two
sequences. (Note that you need to select an appropriate length for the DFT to obtain linear
convolution instead of circular convolution).
iv) Verify that $y_1[n]=y_2[n]$.
v) What is the DFT length you used in iii)? In some applications such large DFT sizes may not be
possible due to computational constraints or real-time processing requirements (i.e. you may
not be allowed to wait to store such long sequences before processing). Hence, overlap-add or
overlap-save methods can be used instead. Implement the overlap-add method where you are
allowed to use DFT and IDFT of only length 256. What is the segment size range you can use? Let
the result with overlap-add method be $y_3[n]$ and verify it equals $y_1[n]$ or $y_2[n]$.

14) (MATLAB) In some cases, to calculate the convolution of two sequences, the convolution theorem
of Discrete Fourier Transformation may be preferable instead of the direct application of the
convolution summation. This is due to the smaller number of arithmetic operations (i.e. additions and
multiplications) of the DFT based method compared to the number of arithmetic operations of the
convolution summation. The small number of arithmetic operations of the DFT is achieved by using an
efficient implementation of the DFT, namely FFT algorithm, which will be discussed at the end of the
term.
i) Generate a N1-point sequence, x1[n] and an N2-point filter h[n], by using round(randn(1,N))
command in MATLAB. Select N1=8192 first and then you may try larger values such as N1=2^15
etc. Select N2=100.
ii) Obtain the convolution result y1[n]=conv(x1,h).
iii) Repeat ii) by using the convolution theorem of DFT (note you can calculate dft using fft(x,L)
where L is DFT length you desire) to obtain y2[n] as the result of the convolution of these two
sequences. (Note that you need to select an appropriate length for the DFT to obtain linear
convolution instead of circular convolution).
iv) Verify that y1[n]=y2[n].
v) What is the DFT length you used in iii)? In some applications such large DFT sizes may not be
possible due to computational constraints or real-time processing requirements (i.e. you may
not be allowed to wait to store such long sequences before processing). Hence, overlap-add or
overlap-save methods can be used instead. Implement the overlap-add method where you are
allowed to use DFT and IDFT of only length 256. What is the segment size range you can use? Let
the result with overlap-add method be y3[n] and verify it equals y1[n] or y2[n].

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