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DCT (Discrete Cosine Transform) is an N-input sequence x(n) , 0≤n≤N-1 , as a linear transformation or combination of complex exponentials. As a result, the DFT coefficients are in general, complex even if x(n) is real.
Suppose, we try to find out an orthogonal transformation which has N×N structure that expressed a real sequence x(n) as a linear combination of cosine sequence. We already know that −
And 
This is possible if N point sequence x(n) is real and even. Thus, 
The resulting DFT itself is real and even. These things make it clear that we could possibly device a discrete cosine transform, for any N point real sequence by taking the 2N point DFT of an “Even extension” of sequence.
DCT is, basically, used in image and speech processing. It is also used in compression of images and speech signals.
DCT is defined by,
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FAQs on Discrete Cosine Transform - Signals and Systems - Electronics and Communication Engineering (ECE)
| 1. What is the Discrete Cosine Transform (DCT)? |  |
| 2. How is the DCT different from the Fourier Transform? | |
Ans. The DCT and the Fourier Transform are both mathematical techniques used for analyzing signals and images in the frequency domain. However, the DCT only uses real-valued cosine functions, while the Fourier Transform uses both cosine and sine functions. This makes the DCT more efficient in terms of computation and storage, especially for real-world signals and images.
| 3. What are the applications of the Discrete Cosine Transform? | |
Ans. The Discrete Cosine Transform has various applications in signal processing and image compression. It is widely used in video and audio compression standards such as JPEG (image compression), MP3 (audio compression), and MPEG (video compression). It is also used in various multimedia applications, digital watermarking, and data transmission over noisy channels.
| 4. How does the DCT achieve data compression? | |
Ans. The DCT achieves data compression by transforming the input signal or image into the frequency domain, where it can be represented with fewer coefficients. Since most of the energy in a signal or image is concentrated in a few low-frequency components, the DCT allows for truncating or quantizing the high-frequency components with minimal loss of information. This leads to efficient data representation and storage, resulting in compression.
| 5. Are there different types of the Discrete Cosine Transform? | |
Ans. Yes, there are different types of the Discrete Cosine Transform, such as DCT-I, DCT-II, DCT-III, and DCT-IV. These types differ in the way the cosine functions are defined and the properties they possess. The most commonly used type is DCT-II, also known as the "standard" DCT, which is used in many practical applications like image and video compression.
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