Basics of Signals Princeton University

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2 CHAPTER 2 BASICS OF SIGNALS, of three variables Now if we are also interested in how the temperature evolves. in time the signal f x y z t would be a function of four variables. The word Matlab,0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4 0 45 0 5. Figure 2 1 Someone saying the word Matlab, Examples of signals that we will encounter frequently are audio signals. images and video An audio signal is created by changes in air pressure and. therefore can be represented by a function of time f t with f representing. the air pressure due to the sound at time t An example of an audio signal. of someone saying Matlab is shown in Figure 2 1 A black and white image. can be represented as a function f x y of two variables Here x y denotes a. particular point on the image and the value f x y denotes the brightness or. gray level of the image at that point, An example of a black and white image is shown in Figure 2 2 A video can. be thought of as a sequence of images Hence a black and white video signal can. be represented by a function f x y t of three variables two spatial variables. and time In this case for a fixed t f t represents the still image frame at. time t while for a fixed x y f x y denotes how the brightness at the point. x y changes as a function of time, Three frames of a video of a commercial are shown in Figure 2 3 It turns.
out that color images or video can be represented by a combination of three. intensity images or video respectively as will be discussed later in Chapter. 2 2 Analog and Digital Signals, Often the domain and the range of a signal f x are modeled as continuous. That is the time or spatial coordinate x is allowed to take on arbitrary values. perhaps within some interval and the value of the signal itself is allowed to. take on arbitrary values again within some interval Such signals are called. 2 2 ANALOG AND DIGITAL SIGNALS 3,Figure 2 2 A gray scale image. 0 13 0 14 0 15,Figure 2 3 Video frames from a commercial. 4 CHAPTER 2 BASICS OF SIGNALS, analog signals A continuous model is convenient for some situations but in. other situations it is more convenient to work with digital signals i e signals. that have a discrete often finite domain and range Two other related words. that are often used to describe signals are continuous time and discrete time. referring to signals where the independent variable denotes time and takes on. either a continuous or discrete set of values respectively. Sampled version of temperature graph,temperature,0 2 4 6 8 10 12 14 16 18 20.
Figure 2 4 Sampling an analog signal, Sometimes a signal that starts out as an analog signal needs to be digitized. i e converted to a digital signal The process of digitizing the domain is called. sampling For example if f t denotes temperature as a function of time and. we are interested only in the temperature at 1 second intervals we can sample. f at the times of interest as shown in Figure 2 4, Another example of sampling is shown in Figure 2 5 An original image. f x y is shown together with sampled versions of the image In the sampled. versions of the image the blocks of constant intensity are called pixels and the. gray level is constant within the pixel The gray level value is associated with. the intensity at the center of the pixel But rather than simply showing a small. dot in the center of the pixel the whole pixel is colored with the same gray level. for a more natural appearance of the image The effect of more coarse sampling. can be seen in the various images Actually the so called original image in. Figure 2 5a is also sampled but the sampling is fine enough that we don t notice. any graininess, The process of digitizing the range is called quantization In quantizing a. signal the value f x is only allowed to take on some discrete set of values as. opposed to the variable x taking on discrete values as in sampling. Figure 2 6 shows the original temperature signal f t shown previously in. Figure 2 4 as well various quantized versions of f Figure 2 7 shows the image. from Figure 2 2 and various quantized versions In the quantized versions of the. images the gray levels can take on only some discrete set of values Actually. 2 2 ANALOG AND DIGITAL SIGNALS 5,No subsampling 4 x 4 blocks. 8 x 8 blocks 16 x 16 blocks,Figure 2 5 Sampling an image.
6 CHAPTER 2 BASICS OF SIGNALS,Unquantized signal 32 levels. 0 0 5 1 1 5 0 0 5 1 1 5,16 levels 8 levels,0 0 5 1 1 5 0 0 5 1 1 5. Figure 2 6 Quantized versions of an analog signal, the so called original image is also quantized but because of the resolution. of the printer and limitations of the human visual system a technique known. as halftoning discussed in Chapter XX can be used so that we don t notice. any artifacts due to quantization It is typical in images to let the gray level. take on 256 integer values with 255 being the brightest gray level and 0 the. darkest In Figures 2 7d f there are only 8 4 and 2 gray levels respectively and. quantization artifacts become quite noticeable, Sampling and quantization to digitize a signal seem to throw away much. information about a signal and one might wonder why this is ever done The. main reason is that digital signals are easy to store and process with digital. computers Digital signals also have certain nice properties in terms of robust. ness to noise as we ll discuss in Section XX However there are also situations. in which analog signals are more appropriate As a result there is often a need. for analog to digital conversion and digital to analog conversion also written. A D and D A conversion In digitizing signals one would also like to know. how much information is lost by sampling and quantization and how best to. do these operations The theory for sampling is clean and elegant while the. theory for quantization is more difficult It turns out that choices for sampling. rates and number of quantization levels also depend to a large extent on system. and user requirements For example in black and white images 256 gray levels. is adequate for human viewing much more than 256 would be overkill while. much less would lead to objectionable artifacts We defer a more detailed con. sideration of sampling and quantization until Chapter XX after we have covered. 2 2 ANALOG AND DIGITAL SIGNALS 7,256 levels 32 levels 16 levels.
8 levels 4 levels 2 levels, Figure 2 7 Quantized versions of a gray scale image. 8 CHAPTER 2 BASICS OF SIGNALS,some additional background material. Before discussing some basic operations on signals we describe a fairly com. mon notational convention which we will also follow Continuous time signals. will be denoted using parentheses such as x t while discrete time signals will. use brackets such as x n This convention also applies even if the indepen. dent variable represents something other than time That is y u denotes a. signal where the domain is continuous while y k indicates a discrete domain. whether or not the independent variables u and k refer to time Often the letters. i j k l m n are used to denote a discrete independent variable. 2 3 Some Basic Signal Operations, In addition to the obvious operations of adding or multiplying two signals and. differentiating or integrating a signal certain other simple operations are quite. common in signal processing We give a brief description of some of these here. The original signal is denoted by x t,Original signal f x. 8 6 4 2 0 2 4 6 8,Amplitude shifted signal f x 1 5.
8 6 4 2 0 2 4 6 8,Time shifted signal f x 3,8 6 4 2 0 2 4 6 8. Figure 2 8 Amplitude and time shifted versions of a signal. The signal a x t where a is some number is just adding a constant signal. to x t and simply shifts the range or amplitude of the signal by the amount. a A somewhat different operation is obtained when one shifts the domain of. the signal Namely the signal x t t0 is a time shift of the original signal x t. by the amount t0 It s like a delayed version of the original signal Figure 2 8. shows amplitude and time shifted versions of a signal. 2 3 SOME BASIC SIGNAL OPERATIONS 9,A periodic signal. 0 20 40 60 80 100 120 140 160 180 200,Figure 2 9 A periodic signal. For some signals appropriate time shifts can leave the signal unchanged. Formally a signal is said to be periodic with period P if x t P x t for all. t That is the signal simply repeats itself every P seconds Figure 2 9 shows an. example of a periodic signal, Amplitude scaling a signal to get ax t is simply multiplying x t with a. constant signal a However a rather different operation is obtained when one. scales the time domain Namely the signal x at is like the original signal. but with the time axis compressed or stretched depending on whether a 1 or. a 1 Of course if a 1 the signal is unchanged Figure 2 10 shows the effects. of amplitude and time scaling For negative values of a the signal is flipped or. reflected about the range axis in addition to any compression or stretching. In particular if a 1 the signal is reflected about the range axis but there. is no stretching or compression For some functions the reflection about the. range axis leaves the function unchanged that is the signal is symmetric about. the range axis Formally the property required for this is x t x t for all. t Such functions are called even A related notion is that of an odd function. for which x t x t These functions are said to be symmetric about the. origin meaning that they remain unchanged if they are first reflected about. the range axis and then reflected about the domain axis Figure 2 11 shows. examples of an even function and an odd function, The signal x y t is called the composition of the two functions x and.
y For each t it denotes the operation of taking the value y t and evaluating. x at the time y t Of course we can get a very different result if we reverse. the order and consider y x t, One other operation that is extremely useful is known as convolution We. will defer a description of this operation until Section XX. 10 CHAPTER 2 BASICS OF SIGNALS,Original signal f x. 10 5 0 5 10,Amplitude scaled signal 2f x,10 5 0 5 10. Time scaled signal f x 2,10 5 0 5 10,Time scaled signal f 2x. 10 5 0 5 10, Figure 2 10 Amplitude and time scaled versions of a signal.
2 4 NOISE 11,An even function,50 40 30 20 10 0 10 20 30 40 50. An odd function,50 40 30 20 10 0 10 20 30 40 50,Figure 2 11 Examples of even and odd functions. In many applications desired signals are subject to various types of degradations. These degradations can arise from a variety of sources such as limitations of the. sensing device random and or unmodeled fluctuations of underlying physical. processes or environmental conditions during sensing transmission reception. or storage of the data The term noise is typically used to describe a wide range. of degradations, It is often useful to try and model certain properties of the noise One. widely used model is to assume that the original desired signal is corrupted. by additive noise that is by adding another unwanted signal Of course if. we knew the noise signal that was added we could simply subtract it off to. get back the original signal and the noise would no longer be an issue Unfor. tunately we usually do not have such detailed knowledge of the noise signal. More realistically we might know or assume that the noise satisfies certain. properties without knowing the exact values of the noise signal itself It is very. common to model the noise as random and assume that we know something. about the distribution of the noise For example we might assume that the. noise is randomly uniformly distributed over some interval or that it has a. Gaussian normal distribution with a known mean and variance Even this. minimal type of knowledge can be extremely useful as we will see later. Robustness to effects of noise can be a major design consideration for certain. systems This can be one reason why for many applications a digital system. 12 CHAPTER 2 BASICS OF SIGNALS,Original signal,0 0 5 1 1 5. Noisy signal,0 0 5 1 1 5,Figure 2 12 Adding noise to an analog signal.
Original signal quantized to 8 levels,0 0 5 1 1 5,Noisy quantized signal. 0 0 5 1 1 5,Original signal recovered by requantization. 0 0 5 1 1 5,Figure 2 13 Adding noise to a quantized signal. 2 5 SOME COMMON SIGNALS 13, might be preferred over an analog one Of course the power of digital computing. is also a key reason for the prevalence of digital systems and robustness to noise. is one factor that makes digital computing so reliable Figures 2 12 and 2 13. illustrate the effect of adding noise on an analog signal and a quantized although. still continuous time signal Without further knowledge of signal and noise. characteristics the noise cannot be removed from an analog signal since any. possible value could be a valid value for the signal On the other hand if we. know the original signal is quantized so it takes on only a discrete set of values. then depending on the noise level it may be possible to remove much of the. noise by simply re quantizing the noisy signal This process simply maps the. observed signal values to one of the possible original levels for example by. selecting the closest level,2 5 Some Common Signals.
Here we briefly define some signals that we will commonly encounter Perhaps. Basics of Signals 2 1 What are Signals As mentioned in Chapter XX a system designed to perform a particular task often uses measurements obtained from the environment and or inputs from a user These in turn may be converted into other forms The physical variables of interest are generally called signals In an electrical system the physical variables of interest might be a voltage

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