# Digital Signal Processing

### Topic: Digital techniques for analyzing analog signals.

### Digital Signal Processing

Digital signal processing(DSP) is concerned with the representation of thesignalsby a sequence of numbers or symbols and the processing of these signals. Digital signal processing andanalog signal processingare subfields ofsignal processing. DSP includes subfields like:audioandspeech signal processing, sonar and radar signal processing, sensor array processing, spectral estimation, statistical signal processing,digital image processing, signal processing for communications, biomedical signal processing, seismic data processing, etc.

Since the goal of DSP is usually to measure or filter continuous real-world analog signals, the first step is usually to convert the signal from an analog to a digital form, by using ananalog to digital converter. Often, the required output signal is another analog output signal, which requires adigital to analog converter. Even if this process is more complex than analog processing and has adiscrete value range, the stability of digital signal processing thanks toerror detection and correctionand being less vulnerable tonoisemakes it advantageous over analog signal processing for many, though not all, applications.

Today there are additional technologies used for digital signal processing including more powerful general purposemicroprocessors,field-programmable gate arrays(FPGAs),digital signal controllers(mostly for industrial apps such as motor control), andstream processors, among others.

### DSP Domains

In DSP, engineers usually study digital signals in one of the following domains:time domain(one-dimensional signals), spatial domain (multidimensional signals), they choose the domain in which to process a signal by making an informed guess (or by trying different possibilities) as to which domain best represents the essential characteristics of the signal. A sequence of samples from a measuring device produces a time or spatial domain representation, whereas adiscrete Fourier transformproduces the frequency domain information that is thefrequency spectrum. Autocorrelation is defined as thecross-correlationof the signal with itself over varying intervals of time or space.

### Signal sampling

With the increasing use ofcomputersthe usage of and need for digital signal processing has increased. In order to use an analog signal on a computer it must be digitized with ananalog to digital converter(ADC). In the discretization stage, the space of signals is partitioned intoequivalence classesand quantization is carried out by replacing the signal with representative signal of the corresponding equivalence class. In the quantization stage the representative signal values are approximated by values from a finite set.

Sampling theoremstates that a signal can be exactly reconstructed from its samples if thesampling frequencyis greater than twice the highest frequency of the signal. In practice, the sampling frequency is often significantly more than twice the required bandwidth.

Adigital to analog converter(DAC) is used to convert the digital signal back to analog. The use of a digital computer is a key ingredient indigital control systems.

### Time & Space domains

The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. Filtering generally consists of some transformation of a number of surrounding samples around the current sample of the input or output signal. There are various ways to characterize filters; for example:

§ A "linear" filter is alinear transformationof input samples; other filters are "non-linear." Linear filters satisfy the superposition condition, i.e. if an input is a weighted linear combination of different signals; the output is an equally weighted linear combination of the corresponding output signals.

§ A "causal" filter uses only previous samples of the input or output signals; while a "non-causal" filter uses future input samples. A non-causal filter can usually be changed into a causal filter by adding a delay to it.

§ A "time-invariant" filter has constant properties over time; other filters such asadaptive filterschange in time.

§ Some filters are "stable", others are "unstable". A stable filter produces an output that converges to a constant value with time, or remains bounded within a finite interval. An unstable filter can produce an output that grows without bounds, with bounded or even zero input.

§ A "finite impulse response" (FIR) filter uses only the input signal, while an "infinite impulse response" filter (IIR) uses both the input signal and previous samples of the output signal. FIR filters are always stable, while IIR filters may be unstable.

Most filters can be described in Z-domain (a superset of the frequency domain) by theirtransfer functions. A filter may also be described as adifference equation, a collection ofzeroesand polesor, if it is an FIR filter, animpulse responseorstep response. The output of an FIR filter to any given input may be calculated byconvolvingthe input signal with theimpulse response.

### Frequency Domain

Signals are converted from time or space domain to the frequency domain usually through theFourier transform. The Fourier transform converts the signal information to a magnitude and phase component of each frequency. Often the Fourier transform is converted to the power spectrum, which is the magnitude of each frequency component squared.

The most common purpose for analysis of signals in the frequency domain is analysis of signal properties. The engineer can study the spectrum to determine which frequencies are present in the input signal and which are missing.

Filtering, particularly in non real-time work can also be achieved by converting to the frequency domain, applying the filter and then converting back to the time domain. This is a fast, O (n log n) operation, and can give essentially any filter shape including excellent approximations tobrick wall filters.

There are some commonly used frequency domain transformations. For example, thecestrumconverts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform. This emphasizes the frequency components with smaller magnitude while retaining the order of magnitudes of frequency components.

Frequency domain analysis is also calledspectrum-orspectral analysis.

### Applications

The main applications of DSP areaudio signal processing,audio compression,digital image processing of the video compression,speech processing,speech recognition,communications,RADAR,SONAR, seismology, and biomedicine. Specific examples are speech compressionand transmission in digitalmobile phones, room matching equalizationof sound in Hifi andsoundreinforcementapplications,weatherforecasting,economicforecasting,seismicdata processing, analysis and control ofindustrial processes, computer-generated animationsinmovies ,In process of medical imaginglikeCATscans&MRI,MP3compression,image manipulation, high fidelity loudspeaker crossovers and equalization, andaudio effectsfor use withelectric guitaramplifiers.

### Implementation

Digital signal processing is often implemented using specialized microprocessors. These often process data usingfixed-point arithmetic, although some versions are available which usefloating point arithmeticand are more powerful. For faster applications FPGAs might be used. Beginning in 2007, multicourse implementations of DSPs have started to emerge from companies includingfree scaleandStream Processors, Inc. For faster applications with vast usage,ASICsmight be designed specifically. For slow applications, a traditional slower processor such as a microcontroller may be adequate.

Techniques

§ Bilinear transform

§ Discrete Fourier transform

§ Discrete-time Fourier transform

§ Filter design

§ LTI system theory

§ Minimum phase

§ Transfer function

§ Z-transform

§ Goertzel algorithm

§ s-plane

Related fields

§ Analog signal processing

§ Automatic control

§ Computer Engineering

§ Computer Science

§ Data compression

§ Electrical engineering

§ Information theory

§ Telecommunication

### Bilinear Transformation

Thebilinear transformis used indigital signal processingand discrete-timecontrol theoryto transform continuous-time system representations to discrete-time and vice versa. The bilinear transform is aconformal mapping, often used to convert atransfer function H_a(s) \ of alinear,time-invariant(LTI) filter in thecontinuous-time domain (often called ananalog filter) to a transfer function H_d(z) \ of a linear, shift-invariant filter in thediscrete-time domain (often called adigital filteralthough there are analog filters constructed withcharge-coupled devicesthat are filters). It positions function that is an exact mapping of the z-plane to the s-plane. When theLaplace transformis performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayedunit impulse), the result is precisely theZ transformof the discrete-time sequence with the substitution of

\begin{align} z &= e^{sT} \\ &= \frac{e^{sT/2}}{e^{-sT/2}} \\ &\approx \frac{1 + s T / 2}{1 - s T / 2} \end{align}

Where T \ is thesample time(the reciprocal of thesampling frequency) of the discrete-time filter. The above bilinear approximation can be solved for s \ or a similar approximation for s = (1/T) \ln(z) \ \ can be performed.

The inverse of this mapping is

\begin{align} s &= \frac{1}{T} \ln(z) \\ &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5 + \frac{1}{7} \left( \frac{z-1}{z+1} \right)^7 + \cdots \right] \\ &\approx \frac{2}{T} \frac{z - 1}{z + 1} \\ &\approx \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}} \end{align}

The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function,

s \leftarrow \frac{2}{T} \frac{z - 1}{z + 1}.

The bilinear transform is a special case of aconformal mapping, defined as Stability and minimum-phase property preserved. A continuous-time causal filter isstableif thepolesof its transfer function fall in the left half of thecomplexs-plane. A discrete-time causal filter is stable if the poles of its transfer function fall inside theunit circlein thecomplex z-plane. Likewise, a continuous-time filter isminimum-phaseif thezerosof its transfer function fall in the left half of the complex s-plane. A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase.

As an example take a simplelow-passRC filter. This continuous-time filter has a transfer function

\begin{align} H_a(s) &= \frac{1/sC}{R+1/sC} \\ &= \frac{1}{1 + RC s}. \end{align}

If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting forsthe formula above; after some reworking.

= H_a \left( \frac{2}{T} \cdot \frac{\left(e^{j \omega T/2} - e^{-j \omega T/2}\right)}{\left(e^{j \omega T/2} + e^{-j \omega T/2 }\right)}\right) \

H_d(z) = H_a \left( \frac{2}{T} \frac{z-1}{z+1}\right) \

H_d(e^{ j \omega T}) \

= H_a \left( \frac{2}{T} \frac{e^{ j \omega T} - 1}{e^{ j \omega T} + 1}\right) \

### Filter design

Filter designis the process of designing a filter, which is a shift-invariant filter, which satisfies a set of requirements, some of which are contradictory. A filter can be designed by process optimization problem where each requirement contributes with a term to an error function which should be minimized. Certain parts of the design process can be automated, but normally an experienced electrical engineer is needed to get a good result.

Typical design requirement

Typical requirements which are considered in the design process are

§ The filter should have a specificfrequency response

§ The filter should have a specificimpulse response

§ The filter should becausal

§ The filter should be stable

§ The filter should be localized

§ The computational complexity of the filter should be low

§ The filter should be implemented in a particular hardware or software.

### The frequency function

Typical examples of frequency function are

§ Alow-pass filteris used to cut unwanted high-frequency signals.

§ Ahigh-pass filterpasses high frequencies fairly well; it is helpful as a filter to cut any unwanted low frequency components.

§ Aband-pass filterpasses a limited range of frequencies.

§ Aband-stop filterpasses frequencies above and below a certain range. A very narrow band-stop filter is known as a notch filter.

An all-pass filter passes through all frequencies unchanged, but changes the phase of the signal. This is a filter commonly used inphaser effects.

An important parameteris the requiredfrequency response. In particular, the steepness and complexity of the response curve is a deciding factor for the filter order and feasibility.

A first orderrecursivefilter will only have a single frequency-dependent component. This means that theslopeof the frequency response is limited to 6dBperoctave. For many purposes, this is not sufficient. To achieve steeper slopes, higher order filters are required.

### TECHNIQUES-DETAILS:

The development of DSP microprocessors has made DSP the method of choice in recent years. Unfortunately, real time DSP requires relatively expensive DSP microprocessors.

If the operations needed to transfer the data to and from the processor are included, it turns out that the processor must perform faster than 400 Mflops if programmed in assembler language and faster than 600 Mflops if programmed in a high level language such as C.

### Details

In this each frame of the imaging video signal, in which the pixels appear in row order, is filtered by an analog filter, transposed to column order, filtered again by an analog filter, and transposed back to row order to give the output video signal. Typically, the two analog filters are identical. It has very low noise to signal ratio.

### THE PREFERRED EMBODIMENTS IN DETAIL

The present invention is of a method and device for processing analog signals, such as video signals, that represent data acquired in a two dimensional field. Specifically, the present invention can be used to filter video signals in real time.

The principles and operation of analog according to the present invention may be better understood with reference to the drawings and the accompanying description.

Pipelined bus structure is used for processing successive frames of video signals according to this method. The double solid lines represent the flow of transfers the resulting digital signals to communications lines 14. D/A converter 26 receive digital signals from communications lines 10 and transform them to analog signals.

The operation of the device of is best understood from a narrative of signal flow through the device of during the processing of several successive frames of video signals.

First video frame, stored in memory bank 36 in digital form and in row order, is read from memory bank 36 and sent to D/A converter 26, where the first video frame is converted to an output analog signal.

All this is done after the receipt of the third input video frame by filter 40 and before the receipt of a fourth input video frame by filter 40.

The processing imposes a uniform delay of two frame durations on the signal. Modern analog video amplifiers impose delays of on the order of 2 nanoseconds on the incoming signals. This is about 4% of the duration of one standard analog video pixel, and therefore is negligible. The read-write cycles of modern memories are on the order of 8 nanoseconds, also considerably shorter than the duration of one standard analog video pixel.

In an alternative embodiment of the present invention, the fact is exploited that once a memory location has been read from during a given frame duration, the data that was stored in that memory location is no longer needed and may be written over. It has, only two memory banks in this embodiment. In this embodiment, during the duration of any one video frame, as twice-filtered row order data are read from one of the memory units.

The results of video processing, using the prototype system mentioned above. This method has many potential applications in microscopy, military target acquisition, surface defect detection, and medical surgery. In the latter application, the present invention can enable a physician at a remote location to see the textures and tiny details of a surgical patient's body tissues.

Conclusion: This term paper describes the various techniques for analyzing the analog signals. I have made my full effort to cover each every technique & explain them in detail.

Digital signal processing also covered in this paper along with a new invention which is recently used for signal processing.

### References:

1) Google search engine.

2) Wikipedia

3) e-book on techniques of dsp transformations.

4) Analog communications by J.S.Katre.

5) Professor J.S. Dhillon (communications)