# DCT Implementation

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**Part 1**

In the box below, implement a function `dct(x)`

, which computes the DCT
coefficients associated with a given input sequence `x`

(given as a NumPy
array). It should return another NumPy array containing the DCT coefficients.

Since the DCT results should be purely real, your function should return a
NumPy array of floats, *not* complex numbers (note that you can find the real
part of an array `x`

by doing `x.real`

).

**Implement this function directly from the formula, without using a
DFT.**

**Part 2**

In the box below, implement a function `dct(x)`

, which computes the DCT
coefficients associated with a given input sequence `x`

(given as a NumPy
array). It should return another NumPy array containing the DCT coefficients.

Since the DCT results should be purely real, your function should return a
NumPy array of floats, *not* complex numbers (note that you can find the real
part of an array `x`

by doing `x.real`

).

**Implement this function using a DFT, not by direct translation of the formula.**

**Part 3: 2D**

In the box below, implement a function `dct2(x)`

, which computes the DCT
coefficients associated with a given input array `x`

(given as a NumPy array).
Your function should return another NumPy array containing the DCT
coefficients, where `out[0,0]`

corresponds to the DC term. Note that, as with
the DFT, we can break this transform down into two smaller 1-D transforms.

You may assume a working copy of the `dct`

function is available for you in the
box below (i.e., you do not need to paste your definition of `dct`

below).