HW4- Professor Mintchev

ECE-210-B HW4

1. Professor Mintchev has just assigned you 20 tedious Gram Schmidt Orthonormalization problems! Luckily, you are a master of MATLAB so you
decide to build a function which can handle all of them for you in short
time.
(a) Create a function called gram_schmidt. The input to the function
should be a 2D array, each column of which is a vector in the original linearly independent set of vectors. Implement GS to create an
orthonormal set of vectors from these. Store them as columns in an
output matrix, similar to the input format. Feel free to use the norm
function if needed.
(b) After you’ve created this function, you’d like a way to test if this
works. Create another function called is_orthonormal which has a
single 2D array as the input. The function should return 1 (or true)
if all columns are orthonormal and 0 (or false) otherwise.
Be careful with this – direct floating-point equality comparison is
a bad idea. Instead apply a threshold to the difference of the two
numbers like so: if |x − xˆ| >  then . . . The eps function might be
useful here. You can add a nice big fudge factor to make the tolerance
big enough that it works, just don’t make it huge.
(c) Finally, we would like to estimate another vector as a linear combination of these orthonormal vectors. (Project the vector on to
the space of the orthonormal vectors.) Implement a function called
ortho_proj which takes a (column) vector to be estimated and an array of orthonormal columns as arguments and outputs the estimated
vector.
(d) Test all of the above functions on some random complex vectors.
(Use rand to make a random vector.) First test the case where there
are more elements in each vector than the number of vectors. Then
test the case where the number of vectors is equal to the number of
elements in a vector. Compare the errors. (You can use the Euclidean
distance as a measure of error.)
1
(e) Uniformly sample sin(x) on [0, 2π] with 1000 points. Also generate
five Gaussians by sampling
1
√
2πσ2
exp
−(x − µ)
2
σ
2
over the same [0, 2π] interval, with σ = 1 and µ ∈ {0, π/2, π, 3π/2, 2π}.
Consider using ndgrid or broadcasting for compact code. Plot the
sinusoid and Gaussians on the same plot. Give axis labels and a title.
Use gram_schmidt to create an orthonormal set of vectors from the
Gaussians. Use ortho_proj to estimate the sinusoid from that set
of vectors.
Create a 2 × 1 subplot (using subplot or tiledlayout). Plot the
sinusoid and the estimated sinusoid together on the upper plot. Plot
the orthonormal basis functions on the lower plot. Give all plots
proper labels and titles.
(f) (Optional) For the first part, you could assume that the input vectors
were a linearly independent set. This simplifies the code a little bit.
How would you deal with a linearly dependent set? Implement it and
test on a linearly dependent set.
(g) (Optional) If you haven’t already done so already, you might find
it neater to use the ortho_proj function in your implementation of
gram_schmidt.
2