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# idct() - Signal Processing

```The inverse
discrete cosine transform reconstructs a sequence from its discrete
cosine transform (DCT) coefficients. The idct function
is the inverse of the dct function.x = idct(y) returns
the inverse discrete cosine transform of yx(n)=∑k=1Nw(k)y(k)cos(π(2n−1)(k−1)2N), n=1,2,…Nwherew(k)={1N,k=12N,2≤k≤Nand N = length(x), which is the same as length(y).
The series is indexed from n = 1 and k = 1 instead of the usual n = 0 and k = 0 because MATLAB® vectors
run from 1 to N instead of from
0 to N-1.x = idct(y,n) appends
zeros or truncates the vector y to length n before
transforming.If y is a matrix, idct transforms
its columns.```

### Syntax

`x = idct(y)x = idct(y,n)`

### Example

```Signal Reconstruction via Inverse Discrete Cosine TransformOpen This Example
Generate a signal that consists of a 25 Hz sinusoid sampled at 1000 Hz for 1 second. The sinusoid is embedded in white Gaussian noise with variance 0.01.
rng('default')

Fs = 1000;
t = 0:1/Fs:1-1/Fs;
x = sin(2*pi*25*t) + randn(size(t))/10;
Compute the discrete cosine transform of the sequence. Determine how many of the 1000 DCT coefficients are significant, that is, greater than 1.y = dct(x);

sigcoeff = abs(y)>=1;

howmany = sum(sigcoeff)

howmany =

17

Reconstruct the signal using only the significant components.y(~sigcoeff) = 0;

z = idct(y);
Plot the original and reconstructed signals.subplot(2,1,1)
plot(t,x)
yl = ylim;
title('Original')

subplot(2,1,2)
plot(t,z)
ylim(yl)
title('Reconstructed')

Related ExamplesDCT for Speech Signal Compression```