This course covers various topics related to transforms and discrete-time systems. The transforms include the Discrete-time Fourier Transform (DTFT), which converts a discrete-time signal into a continuous-frequency spectrum; the Discrete Fourier Transform (DFT), which approximates the DTFT by sampling the frequency domain; the Discrete Cosine Transform (DCT), which is a real-valued transform that is widely used in signal compression; and the Modified DCT (MDCT), which is an extension of the DCT that reduces the blocking artifacts in audio and image coding. The discrete-time systems include linear and circular convolution, which are methods of combining two signals to produce a third one; overlap add and overlap save, which are techniques for implementing fast convolution using the DFT; stability triangle, which is a graphical tool for analyzing the stability of linear time-invariant systems; allpass filters, which are filters that have a constant magnitude response but a variable phase response; group delay, which is a measure of the phase distortion introduced by a filter; and minimum phase systems, which are systems that have the smallest possible group delay for a given magnitude response. The course also covers topics related to FIR and IIR filter design and implementation. FIR filters are filters that have a finite impulse response and can be designed using windowing methods or the remez exchange algorithm for optimal approximation of a desired frequency response. IIR filters are filters that have an infinite impulse response and can be designed using bilinear transformation, which maps analog filters to digital ones; spectral transformations, which modify the frequency characteristics of existing filters; or optimal filter design methods, such as minimax or least-squares. Filter implementation topics include coefficient quantisation, which is the process of rounding or truncating the filter coefficients to fit a finite word length; and lattice filters, which are structures that use only delay elements and multipliers to implement recursive filters. Finally, the course introduces multirate signal processing techniques, such as sample rate conversion, which changes the sampling rate of a signal; polyphase filters, which decompose a signal into subbands with different sampling rates; Farrow filters, which are variable fractional delay filters that can interpolate between samples; and subband processing, which is a method of processing signals in different frequency bands separately.
One of the applications of multirate signal processing is in audio coding, which is the process of compressing and decompressing digital audio signals. Audio coding techniques aim to reduce the bit rate of the audio signal while preserving the perceptual quality as much as possible. One of the common methods of audio coding is transform coding, which uses a transform such as the DCT or the MDCT to convert the time-domain signal into a frequency-domain representation. The transform coefficients are then quantized and encoded using entropy coding techniques such as Huffman coding or arithmetic coding. The quantization and encoding steps exploit the psychoacoustic properties of human hearing, such as masking and critical bands, to discard or reduce the information that is not perceptually important.
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Another application of multirate signal processing is in image compression, which is the process of reducing the size of digital images without compromising the visual quality too much. Image compression techniques can be classified into two categories: lossless and lossy. Lossless compression methods preserve the exact pixel values of the original image and can be used for applications that require high fidelity, such as medical imaging or archival purposes. Lossy compression methods introduce some distortion or artifacts in the reconstructed image and can be used for applications that are more tolerant of quality degradation, such as web browsing or video streaming. One of the popular lossy compression methods is JPEG, which uses a variant of the DCT to transform blocks of pixels into frequency coefficients. The coefficients are then quantized and encoded using a combination of run-length coding and Huffman coding. The quantization step exploits the fact that human vision is more sensitive to low-frequency components than high-frequency ones, and thus allocates more bits to the former than the latter. 0efd9a6b88