DSPTricks A Set of Macros for Digital Signal Processing Plots

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1 DSPTricks A Set of Macros for Digital Signal Processing Plots Paolo Prandoni The package DSPTricks is a set of L A TEX macros for plotting the kind of graphs and figures that are usually employed in digital signal processing publications 1 ; the package relies on PSTricks[1, 2] to generate its graphic output. The basic DSPTricks plot is a boxed chart displaying a discrete-time or a continuoustime signal, or a superposition of both; discrete-time signals are plotted using the lollipop formalism while continuous-time functions are rendered as smooth curves. Other types of plots that commonly occur in the signal processing literature, and for which DSPTricks offers macros, are frequency plots and pole-zero plots. The companion package DSPFunctions defines some signals commonly used in basic signal processing in terms of PostScript primitives, while the package DSPBlocks provides a set of macros to design simple signal processing block diagrams. 1 Drawing Signals Signal plots in DSPTricks are defined by a Cartesian grid enclosed by a box; there are three fundamental types of plots: discrete-time plots, continuous-time plots, frequency-domain plots; discrete- and continuous-time plots can be mixed, whereas frequency-domain plots involve a re-labeling of the horizontal axis in trigonometric units. 1.1 The ÔÈÐÓØ environment ÔÈÐÓØ Data plots are defined by the ÔÈÐÓØ environment as Òß ÔÈÐÓØÐ options ß xmin, xmax Ðß ymin, ymax Ð... Ò ß ÔÈÐÓØÐ This sets up a data plot with the horizontal axis spanning the xmin - xmax interval and with the vertical axis spanning the ymin - ymax interval. The following options are available for all data plots: Û Ø = dim : width of the plot (using any units) 1 The original macros have been written by the author while working on the manuscript for[6]. 1

2 ÔÏ ÔÀ Ø= dim : height of the plot. Width and height specify the size of the active plot area, i.e., of the boxed region of the cartesian plane specified by the x and y ranges for the plot. This is possibly augmented by the space required by the optional labels and axis marks. You can set the default size for a plot by setting the ÔÏ and ÔÀ lengths at the beginning of your document ÜØÝÔ = Ø Ñ Ö Õ plot : type of plot: time domain (default) or digital frequency ÜØ = ÙØÓ Ù ØÓÑ ÒÓÒ step : labeling of the horizontal axis ÝØ = ÙØÓ Ù ØÓÑ ÒÓÒ step : labeling of the vertical axis. When the option specifies a numeric value step, that will be the spacing between two consecutive ticks on the axis 2. When ÙØÓ is selected, the spacing will be computed automatically as a function of the axis range. When ÒÓÒ is selected, no ticks will be drawn. When Ù ØÓÑ is selected, no ticks will be drawn but the plot will include the appropriate spacing for ticks to be drawn later via the Ô Ù ØÓÑÌ macro. Ô= gap : extra space (in horizontal units) to the left and the right of the x -axis range. Useful in discrete-time plots not to have stems overlapping the plot s frame. By default, it s automatically determined as a function of the range; use a value of zero to eliminate the side gap. ÜÓÙØ=ØÖÙ Ð : normally, ticks and tick labels for the horizontal axis are placed on the axis, which may be inside the box; set this option to ØÖÙ if you want to place the ticks on the lower edge of the box in all cases. ÒØ = ØÖÙ Ð : x-axis ticks are normally extending downwards; by setting this option to ØÖÙ ticks will be pointing upwards, i.e. they will be inside the plot box even when the x-axis coincides with the bottom of the box. ÜÐ Ð= label : label for the horizontal axis; placed outside the plot box ÝÐ Ð= label : label for the vertical axis; placed outside the plot box, on the left ÖÐ Ð= label : additional label for the vertical axis; placed outside the plot box on the right Within a ÔÈÐÓØ environment you can use the plotting commands described in the next sections, as well as any PSTricks command; in the latter case, the PSTricks values for ÜÙÒ Ø and ÝÙÒ Ø are scaled to the axes (i.e., they correspond to the cartesian values of the plot). Other useful commands for all data plots are the following: Ô Ð Ô ÔÈÐÓØ Ö Ñ Ô Ù ØÓÑÌ in order to make sure that all drawing commands are clipped to the bounding box defined by the box chart, you can enclose them individually in a predefined Ô Ð Ô environment. See section 1.3 for an example. to redraw the framing box (useful to smooth out plots touching the frame) you can issue the command ÔÈÐÓØ Ö Ñ to draw arbitrarily placed ticks (and tick labels) on either axis, use 2 For digital frequency plots, ÜØ has a different meaning; see Section 1.5 for details. 2

3 Ô Ù ØÓÑÌ options ß pos label pos label... Ð where the axis is specified in the options field as either Ü Ü (default) or Ü Ý and where the argument is a list of space-separated coordinate-label pairs. If you use math mode for the labels, do not use spaces in your formulas since that will confuse the list-parsing macros. ÔÌ ÜØ place a text label anywhere in the plot using the axes coordinates: ÔÌ ÜØ(x, y)ß label Ð 1.2 Plotting Discrete-Time Signals The following commands generate stem (or lollipop ) plots; available options in the commands are all standards PSTricksoptions plus other specialized options when applicable: ÔÌ Ô to plot a set of discrete time points use ÔÌ Ô options ß data Ð where data is a list of space-separated index-value pairs (e.g., values precomputed by an external numerical package). Allowed options are the generic PSTricks plot options. ÔÌ Ô Ø to plot a set of discrete time points use ÔÌ Ô Ø options ß start Ðß data Ð where start is the initial index value and data is a list of space-separated signal values. Allowed options are the generic PSTricks plot options. ÔÌ Ô Ð for large data sets, you can use ÔÌ Ô Ð options ß filename Ð where now filename points to an external text file of space-separated indexvalue pairs. ÔË Ò Ð to plot a discrete-time signal defined in terms of PostScript primitives use ÔË Ò Ð options ß PostScript code Ð ÜÑ Ò ÜÑ Ü ÔË Ò ÐÇÔØ The PostScript code must use the variable Ü as the independent variable; the ÔÈÐÓØ environment sweeps Ü over all integers in the xmin - xmax interval defined for the plot; this can be changed for each individual signal by using the options ÜÑ Ò= m and/or ÜÑ Ü= n. If you use TEX macros in your PS code, make sure you include a space at the end of the macro definition. For instance, use Ò ß¼º Ð. to perform a PostScript initialization sequence before evaluating the signal, use ÔË Ò ÐÇÔØ options ß init Ðß PostScript code Ð where init is a valid PostScript sequence (e.g. ß» ½ ¾ Ð to initialize an array of data). 3

4 For example: 1 Òß ÔÈÐÓØÐß¹ ¾¾Ðß¹½º¾ ½º¾Ð 2 ± ÓÖ ÑÝ ÔÓ Ø Ö ÔØ ÒØ ÖÔÖ Ø Ö Ö Ò Ñ Ü ¼Ü 3 ÔË Ò Ð ÜÑ Ò ßÖ Ò ¾½ Ú ¼º Ù ¾ ÑÙÐÐ 4 ÔÌ Ô Ð Ò ÓÐÓÖ Ö ß ½ ½ ½Ð 5 ÔÌ Ô Ø Ð Ò ÓÐÓÖ ÐÙ ¼ ß¹¾Ðß¹º º Ð 6 Ò ß ÔÈÐÓØÐ produces the following plot: If you are viewing this document in a PostScript viewer, you can see that the random signal is different every time you reload the page, since the taps values are computed on the fly by the PostScript interpreter. 1.3 Plotting Continuous-Time Signals Continuous-time functions can be plotted with the following commands: Ô ÙÒ You can draw a continuous-time signal by using the command Ô ÙÒ options ß PostScript code Ð again, the PostScript code must use Ü as the independent variable; the range for Ü is the xmin - xmax interval and can be controlled for each signal independently via theüñ Ò and ÜÑ Ü options. Ô ÙÒ Ø To plot a smooth function obtained by interpolating a list of space separated time-value pairs use Ô ÙÒ Ø options ß data Ð the interpolation is performed by the PostScript interpreter and can be controlled if necessary by using the appropriate PSTricks options. Ô ÙÒ Ð For a continuous-time smooth interpolation of a pre-computed set of data points, use Ô ÙÒ Ð options ß filename Ð where filename points to a text file containing the data points as a spaceseparated list of abscissae and ordinates. 4

5 Ô Ö To plot one or more Dirac deltas (symbolized by a vertical arrow) use Ô Ö options ß pos value pos value... Ð where the argument is a list of space-separated time-value pairs. In the following example, note the use of the Ô Ð Ô environment when plotting the hyperbola 3 : 1 Òß ÔÈÐÓØÐ ÝØ ½ Ô ¼ ß¼ ½¼Ðß¼ Ð 2 Òß Ô Ð ÔÐ Ô ÙÒß½ Ü Ù Ú Ð Ò ß Ô Ð ÔÐ 3 Ô Ö Ð Ò ÓÐÓÖ Ö ß Ð 4 Ò ß ÔÈÐÓØÐ Plotting Discrete- and Continuous-Time Signals Together In the following examples we mix discrete- and continuous-time signals in the same plot: 1 Òß ÔÈÐÓØÐ ÜØ ½¼ ÝØ ¼º¾ ß¹ ¾¼Ðß¹¼º ½º¾Ð 2 ÒÜßÜ ¼ Õ ß½Ð ßÜ Ê ØÓ Ò Ü ÚÐ Ð Ð 3 ÔË Ò Ð ÜÑ Ü ½¼ ß ÒÜÐ 4 Ô ÙÒ Ð Ò Û Ø ¼º ÔØ ÜÑ Ü ½¼ ß ÒÜÐ 5 Ô ÙÒ Ð Ò ØÝÐ Ð Ò Û Ø ¼º ÔØ ÜÑ Ò ½¼ ß ÒÜÐ 6 Ò ß ÔÈÐÓØÐ Make sure not to leave any blank space in between the beginning and end of the Ô Ð Ô environment, otherwise the axis labels may fall out of alignment. 5

6 1 Òß ÔÈÐÓØÐ Ô ¼º ÝØ ÒÓÒ ß¹ Ðß¹½º¾ ½º¾Ð 2 Ò Ðß ¼º ¾ ÑÙÐ Ê ØÓ Ò Ð 3 ÕÙ ÒØ Þ ß ÙÔ ¼ Ø ß¹¼º Ð ß¼º Ð Ð Ù ØÖÙÒ Ø Ð 4 Ô ÙÒ Ð Ò ÓÐÓÖ Ö Ý Ð Ò Û Ø ¾ÔØ ßÜ ÕÙ ÒØ Þ Ò Ð Ð 5 Ô ÙÒ Ð Ò ØÝÐ ÓØØ Ð Ò Û Ø ½ÔØ ßÜ Ò ÐÐ 6 ÔË Ò ÐßÜ Ò ÐÐ 7 Ò ß ÔÈÐÓØÐ Plotting Digital Spectra Digital frequency 4 plots are set up by setting the option ÜØÝÔ Ö Õ in the ÔÈÐÓØ environment; they are very similar to continuous-time plots, except for the following: the horizontal axis represents angular frequency; its range is specified in normalized units so that, for instance, a range of ß¹½ ½Ð as the first argument to ÔÈÐÓØ indicates the frequency interval[ π,π]. tick labels on the horizontal axis are expressed as integer fractions ofπ; in this sense, the ÜØ parameter, when set to a numeric value, indicates the denominator of said fractions. Ô is always zero in digital frequency plots. ÔÈ Ö Ó Þ All digital spectra are 2π-periodic, hence the[ π, π] interval is sufficient to completely represent the function. However, if you want to explicitly plot the function over a wider interval, it is your responsibility to make the plotted data 2π-periodic; the ÔÈ Ö Ó Þ macro can help you do that, as shown in the examples below. Also, when writing PostScript code, don t forget to scale the x variable appropriately; in particular, PostScript functions of an angle use units in degrees, so you need to multiply Ü by 18 before computing trigonometric functions. 1 Ð Ñ ß¼º Ð 2 Ñ Òß Ð Ñ ½ Ù ÙÔ ÑÙÐ ½ Ð Ñ Ð Ñ ÑÙÐ ± 4 By digital spectrum of a discrete-time sequence x[n] we refer to the Discrete-Time Fourier transform X(e jω )= x[n]e jωn n= 6

7 3 Ü ½ ¼ ÑÙÐ Ó ¾ ÑÙÐ Ð Ñ ÑÙÐ Ù Ú Ð 4 Ô ß Ð Ñ Ü ½ ¼ ÑÙÐ Ò ÑÙÐ ¹½ ÑÙÐ ½ Ð Ñ ± 5 Ü ½ ¼ ÑÙÐ Ó ÑÙÐ Ù Ø Ò ½ ¼ Ú º½ ½ ÑÙÐ Ð 6 7 Òß ÔÈÐÓØÐ ÜØÝÔ Ö Õ ÜØ ÝØ ¼º¾ ± 8 ÝÐ Ð ßËÕÙ Ö Ñ Ò ØÙ À ß ÓÑ Ðµ ¾ Ð ß¹½ ½Ðß¼ ½º½Ð 9 Ô ÙÒß Ñ Ò Ð 1 Ò ß ÔÈÐÓØÐ Òß ÔÈÐÓØÐ ÜØÝÔ Ö Õ ÜØ ÝØ Ù ØÓÑ ± 13 ÝÐ Ð ßÈ Ö Ò µð ß¹½ ½Ðß¹½º ½º Ð 14 Ô ÙÒ ÜÑ Ü ¼ ß Ô Ð 15 Ô ÙÒ ÜÑ Ò ¼ ß¹ Ô ¹½ ÑÙÐÐ 16 Ô Ù ØÓÑÌ Ü Ý ß¹½º ¹ Ô»¾ ¼ ¼ ½º Ô»¾ Ð 17 Ò ß ÔÈÐÓØÐ Square magnitude H(e jω ) π 2π/3 π/3 π/3 2π/3 π π/2 Phase (radians) 2π/3 π/3 π/3 2π/3 π/2 7

8 1 Òß ÔÈÐÓØÐ ÜØÝÔ Ö Õ ÜØ ½ ÝØ ½ ÜÓÙØ ØÖÙ ß¼ ½Ðß¹¼º ½º Ð 2 Ô Ö Ñ ÐÐ ØÝÐ ÚÐ Ò ± 3 Ø ÓÐÓÖ Ð Ø Ö Ý Ø Ò Ð ¾¼ ± 4 Ð Ò ÓÐÓÖ Ð Ø Ö Ý ± 5 ¼ ½º½µ ¼º ¼º µ 6 Ô Ö Ñ ÐÐ ØÝÐ ÚÐ Ò ± 7 Ø ÓÐÓÖ Ð Ø Ö Ý Ø Ò Ð ¾¼ ± 8 Ð Ò ÓÐÓÖ Ð Ø Ö Ý ± 9 ¼º ¼º µ ½ ¹¼º µ 1 Ô Ð Ò Ð Ò Û Ø ¼º ÔØ ¼º ¹º µ ¼º ½º µ 11 Ô Ð Ò Ð Ò Û Ø ¼º ÔØ ¼º ¹º µ ¼º ½º µ 12 Ô ÙÒ Ð Ò ÓÐÓÖ Ö ßÜ º½ ÑÙÐ ¼º ÑÙÐ ½¼ ÜÔ ½ ½ Ü ÚÐ 13 ÔÈÐÓØ Ö Ñ 14 Ô Ù ØÓÑÌ ß¼º ¼º Ô ¼º ¼º Ô Ð 15 Ò ß ÔÈÐÓØÐ 1.4π.6π π The following example shows how to repeat an arbitrary spectral shape over more than one period. First let s define (and plot) a non-trivial spectral shape making sure that the free variable x appears only at the beginning of the PostScript code: 1 ± ØÖ Ò ÙÐ Ö Ô 2 ØÖ ÙÒß ¼º¾ Ù ½ ¼º¾ Ù Ú Ð 3 ± Ô Ö ÓÐ Ô 4 Ô Ö ÙÒß ¼º¾ Ú ÙÔ ÑÙÐ ½ Ü Ù Ð 5 ± ÓÑÔÓ Ø Ô ÙØÓ Ø ¼º Ô µ 6 ÓÑ ÙÒß 7 ÙÔ ÙÔ ÙÔ ÙÔ ± 8 ¹¼º ÐØ ßÔÓÔ ÔÓÔ ÔÓÔ ÔÓÔ ¼Ð ß ± Þ ÖÓ ÓÖ Ü ¹¼º 9 ¼º Ø ßÔÓÔ ÔÓÔ ÔÓÔ ¼ Ð ß ± Þ ÖÓ ÓÖ Ü ¼º 1 ¹¼º¾ ÐØ ßÔÓÔ ØÖ ÙÒ Ð ß ± ØÖ Ò Ð ØÛ Ò 11 ¼º¾ Ø ß ØÖ ÙÒ Ð ± ¹º¾ Ò ¹º 12 ß Ô Ö ÙÒÐ ± Ð Ô Ö ÓÐ 13 Ð Ð± 14 Ð Ð± 15 Ð Ð± 16 Ð Ð Òß ÔÈÐÓØÐ ÜØÝÔ Ö Õ ÝÐ Ð ß ß ÓÑ Ðµ Ð ß¹½ ½Ðß¼ ½º½Ð 19 Ô ÙÒßÜ ÓÑ ÙÒ Ð 2 Ò ß ÔÈÐÓØÐ 8

9 1 X(e jω ) π π/2 π/2 π ÔÈ Ö Ó Þ Now we can periodize the function using the ÔÈ Ö Ó Þ macro; plotting multiple periods becomes as simple as changing the axis range: 1 Òß ÔÈÐÓØÐ ÜØÝÔ Ö Õ ß¹¾ ¾Ðß¼ ½º½Ð 2 Ô ÙÒßÜ ÔÈ Ö Ó Þ ÓÑ ÙÒ Ð 3 Ò ß ÔÈÐÓØÐ 4 5 Òß ÔÈÐÓØÐ ÜØÝÔ Ö Õ ÜØ ½ ß¹ Ðß¼ ½º½Ð 6 Ô ÙÒßÜ ÔÈ Ö Ó Þ ÓÑ ÙÒ Ð 7 Ò ß ÔÈÐÓØÐ 1 2π 3π/2 π π/2 π/2 π 3π/2 2π 1 5π 4π 3π 2π π π 2π 3π 4π 5π 9

10 1.6 Plotting Analog Spectra To plot analog spectra, just set up a plot environment as you would to plot a continuoustime signal, then set the option ÜØ Ù ØÓÑ and place your own frequency labels using Ô Ù ØÓÑÌ as in the example below: 1 Òß ÔÈÐÓØÐ ÜØÝÔ Ö Õ ÜØ Ù ØÓÑ ÜÐ Ð ß Ö Õº ÀÞµÐ ± 2 ÝØ ¾ ÝÐ Ð ß ÇÑ µ Ð ß¹½¼ ½¼Ðß¹½ Ð 3 Ô ÙÒßÜ Ø ß¼Ð ßÜ ¾ Ú ÙÔ ÑÙÐ Ü Ù Ð Ð Ð 4 Ô Ù ØÓÑÌ ß¹ ¹ ÇÑ Æ ¼ ¼ ÇÑ Æ ÇÑ Ð 5 Ò ß ÔÈÐÓØÐ 4 X(jΩ) 2 Ω N Ω N Ω s freq. (Hz) 1.7 Common Signal Shapes and Helper Functions To facilitate the creation of plots that commonly occur in signal processing theory, the package DSPFunctions provides a PostScript implementation for the following set of functions; each macro acts on the free variable x in a plot command (see examples below). Basic Shapes: ÔÊ Øß Ðß Ð computes a rectangular (box) function centered in a and with support 2b, i.e. rect((x a)/b) where 1 if Ü<1/2 rect(x)= otherwise ÔÌÖ ß Ðß Ð computes a triangle function centered in a and with support 2b ÔË Òß Ðß Ð computes the scaled sinc function sinc((x a)/b), where sinc(x)= sin(πx) πx ÔÉÙ ß Ðß Ð computes a quadratic function (inverted parabola) centered in a and with support 2b 1

11 Ô ÜÔ ß Ðß Ð computes the decaying exponential response b ( x a)u[x a] ÔÈÓÖ Ô ß Ðß Ð computes a porkpie hat shape centered in a and with support 2b ÔÊ Ó ß Ðß ÐßÖÐ computes a raised cosine centered in a with cutoff b and rolloff r Discrete Fourier Transform: Ô ÌÅ ßÜ ¼ Ü ½ ººº Ü ßÆ¹½ÐÐ computes the magnitude of the Discrete Fourier transform (DFT) of the provided data points 5 ; the input value should be an integer. N 1 X[k]= x[n]e j 2π N nk Ô ÌÊ ßÜ ¼ Ü ½ ººº Ü ßÆ¹½ÐÐ computes the real part of the DFT n= Ô ÌÁÅßÜ ¼ Ü ½ ººº Ü ßÆ¹½ÐÐ computes the imaginary part of the DFT Notable DTFTs: ÔË ÒËß ÐßÆÐ computes the Discrete-Time Fourier transform (DTFT) of a zero-centered, symmetric 2N + 1-tap rectangular signal: X(e jω sin(ω(2n+ 1)/2) )= sin(ω/2) The parameter a can be used to shift the DTFT to the chosen center frequency. ÔË Ò ß ÐßÆÐ computes the DTFT magnitude of a causal N -tap rectangular signal: X(e jω ) = sin(ω(n/2)) sin(ω/2) The parameter a can be used to shift the DTFT to the chosen center frequency. Frequency Responses: Ô ÁÊÁß ¼ ½ ººº ßÆ¹½ÐÐ computes the (real-valued) frequency response of a zero centered(2n+ 1)-tap Type-I FIR filter with coefficients a N 1, a N 2,..., a 1, a, a 1, a 2,..., a N 1. The coefficient a is the center tap and you need only specify the coefficients from a to a N 1. ÔÌ Åß ¼ ½ ººº ßÆ¹½ÐÐß ½ ¾ ººº ßÅ¹½ÐÐ computes the magnitude response of a generic digital filter defined by the constant-coefficient difference equation: y[n]= a x[n]+...+a N 1 x[n N+ 1] b 1 y[n 1]... b M 1 y[n M+ 1] 5 Please note that the underlying implementation of the macro is not optimized; the computing time will be quadratic in the number of data points. 11

12 For instance: 1 Ù Ô ß Ô ÙÒØ ÓÒ Ð 2 ººº 3 Òß ÔÈÐÓØÐ Ô ½ ß¹¾ ½¼Ðß¹½º¾ ½º¾Ð 4 Ô ÙÒßÜ ÔÊ Øß¹½Ðß½ÐÐ 5 Ô ÙÒßÜ ÔÈÓÖ Ô ß Ðß¾ÐÐ 6 ÔË Ò Ð Ð Ò ÓÐÓÖ Ö Ý ßÜ ÔË Òß¾Ðß Ð ¹½ ÑÙÐÐ 7 Ò ß ÔÈÐÓØÐ We can generate a finite-length signal and plot its DFT magnitude as in this example, where the 32-point input signal is cos(2π/32 (27/5)n): 1 Òß ÔÈÐÓØÐß¼ ½Ðß¼ ½ Ð 2 ÔË Ò ÐÇÔØß» ¼ ½ ½ ß ¼ ¾ Ú º ÑÙÐ ÑÙÐ Ó Ð ÓÖ Ð 3 ßÜ Ô ÌÅ ß ÐÓ ÔÓÔ ÐÐ 4 Ò ß ÔÈÐÓØÐ The magnitude of simple FIR and IIR filter can be graphed easily like so: 1 Òß ÔÈÐÓØÐ ÜØÝÔ Ö Õ ÜÓÙØ ØÖÙ ß¹½ ½Ðß¹¼º ½º Ð 2 Ô ÙÒ Ð Ò ÓÐÓÖ Ö Ý Ð Ò ØÝÐ ßÜ ÔË ÒËß¼Ðß Ð ½ ÚÐ 3 Ô ÙÒßÜ Ô ÁÊÁß ¼º ¼½ ¼º¾ ¾ ¼º½¾ ¾ ¹¼º¼¾½ ¹¼º¼ 4 ¹¼º¼ ¼º¼ Ð Ð 5 Ò ß ÔÈÐÓØÐ 12

13 1 π π/2 π/2 π 1 Òß ÔÈÐÓØÐ ÜØÝÔ Ö Õ ÜØ ß¹½ ½Ðß¼ ½º½Ð 2 ± ¹Ø ÓÖ Ö Ý Ú ÐØ Ö Ü ÑÔÐ 3 ± Ó ÒØ ÖÓÑ Å ØÐ Ù Ò Ý½ ¼º ¼º¾ µ 4 Ô ÙÒßÜ ÔÌ Åß¼º¼¼¼¼¼ ¾ ½½ ¼º¼¼¼¼ ½ ¾¼ ½½½ 5 ¼º¼¼¼¾ ¼ ½½ ¼º¼¼¼ ¼½ ¾ ¼º¼¼¼ ¾ ¾ ¾ 6 ¼º¼¼¼ ¼½ ¾ ¼º¼¼¼¾ ¼ ½½ ¼º¼¼¼¼ ½ ¾¼ ½½½ 7 ¼º¼¼¼¼¼ ¾ ½½ Ðß ½º¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼ ¹ º ¾ ¾¾ ½ 8 ½ º ½¾¾ ¾ ¾¼¾½¼¼ ¹¾ º ½ ¾ ¾ ¾ ¾¾ ¼º ¼ ½¾ 9 ¹¾¾º ¾ ¼ ¾ ½¼º ¼ ¼¼ ½¼ ¹ º¼ ¾ 1 ¼º ¼ ¼ ¼½ ÐÐ 11 Ò ß ÔÈÐÓØÐ 1 π 3π/4 π/2 π/4 π/4 π/2 3π/4 π 2 Drawing Regions of Convergence, Poles and Zeros ÔÈ ÈÐÓØ Pole-zero plots are defined by the environment Òß ÔÈ ÈÐÓØÐ options ß M Ð... Ò ß ÔÈ ÈÐÓØÐ This plots a square section of the complex plane in which both axes span the[ M, M] interval. Options for the plot are: Û Ø = dim : width of the plot 13

14 Ø= dim : height of the plot. Normally, since the range is the same for both the real and the imaginary axis, width and height should be equal. You can therefore specify just one of them and the other will be automatically set. If you explicitly specify both, you will be able to obtain an asymmetric figure. By default, width and height are equal to ÔÀ. ÜØ = ÙØÓ ÒÓÒ d : labeling of the real axis ÝØ = ÙØÓ ÒÓÒ d : labeling of the imaginary axis. When the option specifies a numeric value d, that will be the spacing between two consecutive ticks on the axis. ÙÒ Ø = ØÖÙ Ð : if true, labels the real and imaginary axis with Re and Im respectively. ÖÐ = r : draws a circle centered in z= with radius r ; by default r= 1, so that the unit circle will be drawn; set to zero for no circle. Ð Ð= label : for a circle of radius r, places the selected label text at z = r+ j. By default the label is equal to the value of r. ÖÓ= r : draws a causal region of convergence with radius r. ÒØ ÖÓ= r : draws an anticausal region of convergence with radius r. 2.1 Poles and Zeros ÔÈ To plot a pole or a zero at z= a+ j b use ÔÈ options ß a, b Ð which plots a pole by default; to plot a zero use the option ØÝÔ Þ ÖÓ. To associate a label to the point, use the option Ð Ð text ; if text is ÒÓÒ no label is printed; if text is ÙØÓ (which is the default) the point s coordinates are printed; otherwise the specified text is printed. Finally, you can specify the position of the label using the option ÐÔÓ angle ; by default, the angle s value is 45 degrees. 1 Òß ÔÈ ÈÐÓØÐ Ð Ð ß Ö ¼ Ð ÖÓ ¼º ß½º Ð 2 ÔÈ ÈÓ ÒØ Ð Ð ÒÓÒ ß¼º ¼º Ð 3 ÔÈ ÈÓ ÒØ ØÝÔ Þ ÖÓ Ð Ð ß Ü ½ Ð ÐÔÓ ½ ß¼ ½Ð 4 ÔÈ ÈÓ ÒØ ØÝÔ Þ ÖÓ Ð Ð ß Ü ¼ Ð ÐÔÓ ¼ ß½º¾ ¼º Ð 5 Ò ß ÔÈ ÈÐÓØÐ 14

15 Im x[1] + x[] Re r 3 Block Diagrams Ô ÐÓ Block diagrams rely heavily on PSTricks Ô Ñ ØÖ Ü environment, for which ample documentation is available. To set up a block diagram use the environment Òß Ô ÐÓ Ðß x Ðß y Ð... Ò ß Ô ÐÓ Ð where x and y define the horizontal and vertical spacing of the blocks in the diagram. Predefined functional blocks are listed in the table below and they can be used anywhere a node is required. Nodes are labeled in top-left matrix notation, i.e. the topmost leftmost node is at coordinates(1, 1) and indices increase rightward and downward. Connections between nodes can be drawn using PSTricks standard primitive ÒÐ Ò ; the package defines the following shorthands: ÓÒÒÀÆ ÜØ ÓÒÒÀ ÓÒÒÎ to connect with an arrow a node at(n, m) to its neighboring node at(n, m+ 1) use ÓÒÒÀÆ ÜØß n Ðß m Ð to connect with an arrow a node at(n, m) to a node on the same row at(n, p) use ÓÒÒÀ options ß n Ðß m Ðß p Ðß label Ð, which uses options as line options and label as the label for the connection to connect with an arrow a node at(n, m) to a node on the same column at(q,m) use ÓÒÒÎ options ß n Ðß m Ðß q Ðß label Ð 15

16 function macro output nodes ÔÐ Ø + ÑÙÐ delays Ð Ý z 1 Ð ÝÆß N Ð z N filters ÐØ Öß label Ð H(z) ÐØ ÖÅÙÐØ ß labels Ð multiple lines (use to separate lines) ÐÓÛÔ (you can specify the size of the block, eg ÐÓÛÔ ¾ Ñ ) 16

17 function macro output sampler ÑÔÐ Ö ÑÔÐ Ö Ö Ñ (you can specify the size of the block, eg ÑÔÐ Ö Ö Ñ ¾ Ñ ) interpolator Ò (you can specify the size of the block, eg Ò ¾ Ñ ) upsampler ÙÔ ÑÔß N Ð 3 downsampler Û ÑÔß N Ð 3 17

18 1 Òß Ô ÐÓ Ðßº Ðß½Ð 2 ± Ö Ø ÖÓÛ 3 Ü Ò ² ² ² ÔÐ Ø ² Ð Ý ²± 4 ÔÐ Ø ² Ð Ý ² ÔÐ Ø ² Ð Ý ² ÔÐ Ø ² Ô ß ÑÐ ² ± 5 ² Ð Ý ² 6 ± 7 ± ÓÒ ÖÓÛ 8 ² ² ² ² ²± 9 ² ² ² ² ² Ô ß ÑÐ ² ± 1 ² ² ² ² Ý Ò 11 ± 12 ± ÓÒÒ Ø ÓÒ 13 ÒÐ Ò ß½ ½Ðß½ Ð 14 ÒÐ Ò ß½ Ðß½ Ð 15 ÒÐ Ò ß½ Ðß½ Ð 16 ÒÐ Ò ß½ Ðß½ Ð 17 ÒÐ Ò ß½ Ðß½ ½¼Ð 18 ÒÐ Ò Ð Ò ØÝÐ ÓØØ ß½ ½¼Ðß½ ½¾Ð 19 ÒÐ Ò ß½ ½¾Ðß½ ½ Ð 2 ÒÐ Ò ß½ ½ Ðß½ ½ Ð 21 ÒÐ Ò ß¾ ½¼Ðß¾ ½½Ð 22 ÒÐ Ò Ð Ò ØÝÐ ÓØØ ß¾ ½¼Ðß¾ ½ Ð 23 ÒÐ Ò ß½ Ðß¾ Ð ØÐÔÙØß ¼ Ð 24 ÓÒÒÀß¾Ðß Ðß ÐßÐ 25 ÓÒÒÎß½Ðß Ðß¾Ðß ½ Ð 26 ÓÒÒÀß¾Ðß Ðß ÐßÐ 27 ÓÒÒÎß½Ðß Ðß¾Ðß ¾ Ð 28 ÓÒÒÀß¾Ðß Ðß½¼ÐßÐ 29 ÓÒÒÎß½Ðß½¼Ðß¾Ðß Ð 3 ÓÒÒÀÆ ÜØß¾Ðß½ Ð 31 ÓÒÒÎß½Ðß½ Ðß¾Ðß ßÅ¹½Ð Ð 32 ÓÒÒÀß¾Ðß½ Ðß½ ÐßÐ 33 Ò ß Ô ÐÓ Ð x[n] z 1 z 1 z 1 z 1 b b 1 b 2 b 3 b M y[n] 1 Òß Ô ÐÓ Ðß¾Ðß¼º Ð 2 Ü Øµ ² ÐÓÛÔ ¼º Ñ ² ÑÔÐ Ö Ö Ñ ¼º Ñ ² Ü Ò 3 ÒÐ Ò ß¹ Ðß½ ½Ðß½ ¾Ð 4 ÒÐ Ò ß½ ¾Ðß½ Ð ß Ü ßÄÈÐ Øµ Ð 5 ÒÐ Ò ß¹ Ðß½ Ðß½ Ð 6 Ò ß Ô ÐÓ Ð x(t) x LP (t) x[n] If you need to label nodes for complex connections, you may need to revert to the actual code for the node element, eg: 18

19 1 Òß Ô ÐÓ Ðß¼º Ðß¼Ð 2 ² ² ² Ò Ñ ½ ÐØ Öß À Þµ Ð ² ² ß¹ ÓÑ ÒÐ 3 Øß Ð Øµ ² ÑÔÐ Ö 4 ² Ò Ñ ÑÒÓ ÓØ Ð Ò Û Ø ¾ÔØ ² 5 ² Ò Ñ ÑÒÓ ÖÐ ² ÑÙÐ ² Ò Ñ 6 ± 7 Ô Øß ÖÖÓÛ ¹ Ð 8 Ò Ò Ð Ò Ð ¼ Ò Ð ½ ¼ Ð Ò Û Ø Û Ø ß Ðß ½Ð 9 Ò Ò Ð Ò Ð ¼ Ò Ð ¼ Ð Ò Û Ø Û Ø ß ½Ðß Ð ß Ð 1 Ò Ö Ò Ð ¹ ¼ Ò Ð ¹ ¼ Ð Ò Û Ø Û Ø ß Ðß Ð 11 ÒÐ Ò ß¹Ðß¾ ½Ðß¾ ¾Ð 12 ÒÐ Ò ß¹Ðß¾ ¾Ðß¾ Ð ß Øß Ð Ò Ð 13 ÒÐ Ò ß¾ Ðß¾ Ð ß ØßÐ Ò Ð 14 ÒÐ Ò ß¹Ðß¾ Ðß¾ Ð ß Øß Ð Ò Ð 15 ÒÐ Ò ß½ Ðß¾ Ð 16 Ò ß Ô ÐÓ Ð Ú Ô ß ÜÐ 19 2 Òß Ô ÐÓ Ðß½º¾Ðß¼Ð 21 Ò Ñ ÑÒÓ ÖÐ Ã ÓÛÒ ÖÖÓÛ ² ÐØ ÖßËÐ ÖÐ ² 22 ÐØ Öß Ö Ñ Ð ÖÐ ² 23 Ô ß½ ÜÐ Ô Ö ÓÜß ÜÐß Ñ ÐÐ ØØ ºº¼½½¼¼ ¼½¼½¼ºººÐ 24 Ô Øß ÖÖÓÛ ¹ Ð 25 ÒÐ Ò ß½ ½Ðß½ ¾Ð ß Øß Ð Ò Ð 26 ÒÐ Ò ß½ ¾Ðß½ Ð 27 ÒÐ Ò ß½ Ðß½ Ð 28 Ò ß Ô ÐÓ Ð 29 Ò ÖÖ Ò Ð ¼ Ð Ò Û Ø ½º¾ÔØ Ð Ò Ö ¼º¾ ß Ðß Ð H(z) j e jωc n ŝ[n] ĉ[n] ˆb[n] ŝ(t) + â[n] K Slicer Descrambler

20 References [1] Timothy Van Zandt, PSTricks - PostScript macros for generic TEX, ØØÔ»»ÛÛÛº ØÙ ºÓÖ» ÔÔÐ Ø ÓÒ»ÈËÌÖ, [2] PSTricks Web pages, maintened by Herbert Voß. ØØÔ»»ÛÛÛºÔ ØÖ º [3] Adobe Systems Incorporated, PostScript Language Reference Manual, Addison- Wesley, 3 edition, [4] Bill Casselman Mathematical Illustrations, Cambridge University Press, 25. [5] Michael Mehlich, fp : Fixed point arithmetic for TEX, Ì Æ Ñ ÖÓ»Ð Ø Ü» ÓÒØÖ» Ô. [6] Paolo Prandoni and Martin Vetterli, Signal Processing for Communications, 28, ØØÔ»»ÛÛÛº Ô ÓÑÑºÓÖ 2

Models, Notation, Goals

Models, Notation, Goals Scope Ë ÕÙ Ò Ð Ò ÐÝ Ó ÝÒ Ñ ÑÓ Ð Ü Ô Ö Ñ Ö ² Ñ ¹Ú ÖÝ Ò Ú Ö Ð Ö ÒÙÑ Ö Ð ÔÓ Ö ÓÖ ÔÔÖÓÜ Ñ ÓÒ ß À ÓÖ Ð Ô Ö Ô Ú ß Ë ÑÙÐ ÓÒ Ñ Ó ß ËÑÓÓ Ò ² Ö Ò Ö Ò Ô Ö Ñ Ö ÑÔÐ ß Ã ÖÒ Ð Ñ Ó ÚÓÐÙ ÓÒ Ñ Ó ÓÑ Ò Ô Ö Ð Ð Ö Ò Ð ÓÖ Ñ