ECE5655 FM4 Lab 2. March 8, Problem 1 5. Problem 2 5. Problem 3 7 Part a... 8 Part b... 8 Part c... 8

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1 ECE5655 FM4 Lab 2 March 8, 2016 Contents Linear Algebra with Python 2 A Quick Look at Sympy for Linear Algebra Problem 1 5 Problem 2 5 Problem 3 7 Part a Part b Part c Problem 4: Real-Time Gold Code with LUT and Pulse Shaping 8 PN Code Generation Pulse Shaping and Upsampling by Four Import the Full CA Gold Code Matrix Python Model of the Required C Code Used for Pulse Shaping Support Code and Examples Function that Writes Gold Code Headers Write FIR Header Files Write Raised Cosine and Root Raised Cosine Headers for 4 Samples/Bit Process Data In a CSV File Exported from Analog Discovery 14 Get on with the Problem 4 Lab Write Up In [1]: %pylab inline #%matplotlib qt # for popout plots from future import division # use so 1/2 = 0.5, etc. import ssd import scipy.signal as signal from IPython.display import Image, SVG Populating the interactive namespace from numpy and matplotlib In [103]: pylab.rcparams[ savefig.dpi ] = 100 # default 72 #pylab.rcparams[ figure.figsize ] = (6.0, 4.0) # default (6,4) #%config InlineBackend.figure_formats=[ png ] # default for inline viewing #%config InlineBackend.figure_formats=[ svg ] # SVG inline viewing %config InlineBackend.figure_formats=[ pdf ] # render pdf figs for LaTeX 1

2 Linear Algebra with Python To get started with numerical linear algebra calculations in Python we need to import the linear algebra module from, scipy.linalg. In [9]: import scipy.linalg as linalg We can now create matrices as 2D numpy ndarrays: In [12]: A = array([[1,2,3,4],[5,6,7,8]]) A Out[12]: array([2, 3]) In [13]: AT = A.T AT Out[13]: array([[1, 5], [2, 6], [3, 7], [4, 8]]) In [15]: B = array([[1,2],[4,9]]) BI = linalg.inv(b) In [16]: B Out[16]: array([[1, 2], [4, 9]]) In [17]: BI Out[17]: array([[ 9., -2.], [-4., 1.]]) In [18]: dot(b,bi) Out[18]: array([[ 1., 0.], [ 0., 1.]]) I can now write LaTeX math in a markdown cell to mimic the above numerical calculations: [ ] A = (1) A t = (2) 4 8 With Python sympy, the computer algebra system (CAS) that is part of the Python eco-system, a fully symbolic matrix that is already displayed in LaTeX can be created. 2

3 A Quick Look at Sympy for Linear Algebra In [5]: from IPython.display import display from sympy.interactive import printing printing.init_printing(use_latex= mathjax ) import sympy as sym A, B, C = sym.symbols("a B C") In [8]: A = sym.matrix([[1,2,3,4],[5,6,7,8]]) A Out[8]: [ ] In [9]: A.T Out[9]: In [11]: a,b,c,d = sym.symbols("a b c d") In [13]: # All symbolic matrix B = sym.matrix([[a,b],[c,d]]) B Out[13]: [ ] a b c d In [17]: B.det() Out[17]: A sample plot: ad bc In [44]: n = arange(0,500) x = cos(2*pi*n/500*2.5) plot(n,x) xlabel(r Time Index $n$ ) ylabel(r Amplitude ) title(r Plot of $\cos(2\pi n/500 \times 2.5)$ ) grid(); 3

4 1.0 Plot of cos(2πn/ ) 0.5 Amplitude Time Index n Importing graphics created outside the IPython Notebook: In [46]: Image( coolterm.png,width= 60% ) # Note the image scaling handle to the lower right of the image. Out[46]: 4

5 Problem 1 Develop a C calling C function that implements the numerical calculation using the data type int16 t, where C = A B A = [ a 2 + (a + 1) 2 + (a + 2) (2a 1) 2] B = [ b 2 + (b + 1) 2 + (b + 2) (2b 1) 2] You can perform calculations right in the notebook using Python with the magic %pylab filling the workspace with numpy for array-based mathematics and matplotlib for 2D and 3D interactive graphics. Problem 2 Wolfram Mathematica or WolframAlpha on the Web can be useful too. The images imported in the notebook were captured on my ipad using the WolframAlpha for ipad app: 5

6 In [49]: Image( inv3by3.png,width= 40% ) Out[49]: In [48]: Image( det3by3.png,width= 40% ) Out[48]: 6

7 Some C Code int main(void){ int16_t x = 0; char my_debug[80]; float32_t buffer[4] = {1.0,2.3,3.5,-6.7}; static float32_t AData[2*4] = {1.0f,2.0f,3.0f,4.0f,5.0f,6.0f,7.0f,8.0f}; static float32_t ATData[4*2] = {0.0f,0.0f,0.0f,0.0f,0.0f,0.0f,0.0f,0.0f}; arm_matrix_instance_f32 A = {2,4,AData}; arm_matrix_instance_f32 AT = {4,2,ATData}; Problem 3 In this program you will convert the pseudo-code for a square-root algorithm shown below into C code for float32 t input/output variables. Approximate square root with bisection method INPUT: Argument x, endpoint values a, b, such that a < b OUTPUT: value which differs from sqrt(x) by less than 1 done = 0 a = 0 b = square root of largest possible argument (e.g. ~216). 7

8 c = -1 do { c_old = c c = (a+b)/2 if (c*c == x) { done = 1 } else if (c*c < x) { a = c } else { b = c } } while (!done) && (c!= c_old) return c Part a Code the above square root algorithm in C. Profile you code using the test values 23, 56.5, and Run tests at compiler optimization Level 0 and Level 3. Note: You will need to establish a stopping condition, as the present form is designed for integer math. I suggest modifying the line: if (c*c == x) { to something like if (fabs(c*c - x) <= max_error) } where max-error is intially set to. Realize that this value directly impacts the execution speed, as a smaller error requirement means more iterations are required. See if you can find the accuracy of the standard library square root. Part b Compare the performance of your square root function at O3 with the standard math library function for float (float32 t), using float sqrtf(float x). Tests were run using... Part c Compare the performance of your square root function at O3 to the M4 FPU intrinsic function float32 t sqrtf(float x). Tests were run using... Problem 4: Real-Time Gold Code with LUT and Pulse Shaping Pseudo-random sequences find application in digital communications system. The most common sequences are known as M-sequences, where M stands for maximal length. A Gold Code formed by exclusive ORing two M sequences of the same length but of different phases. For example Gold codes of length 1023 are uniquely assigned to the GPS satellites so that the transmissions from the satellites may share the same frequency spectrum, but be separated by the properties of the Gold codes which make nearly mutually orthogonal. In this problem you start by building an M-sequence generator in C. PN Code Generation Implement as described and test via GPIO pins. You may also want to send output to the terminal and import into Python to plot and verify properties. I will want to see waveforms on the scope and/or logic analyzer. Use the synch pulse output a synch signal for the scope 8

9 Pulse Shaping and Upsampling by Four A prototype of the upsampling can be implemented using Python, right here in the notebook. We will use the actual CA code sequence. This requires importing the entire matrix then just using one row as the bit stream. In [16]: import digitalcom as dc In [35]: len(filt_states) Out[35]: 48 Import the Full CA Gold Code Matrix In [4]: camat = loadtxt( ca1thru37.txt,dtype=int16,unpack=true) In [4]: # Check the size of the matrix camat.shape Out[4]: (37L, 1023L) In [5]: # Check the data type camat.dtype Out[5]: dtype( int16 ) Python Model of the Required C Code Used for Pulse Shaping This code resides in the codec ISR. In [61]: # Pulse shaping filter at 4x oversampling b_src = dc.sqrt_rc_imp(4,0.35) # 0.35 <==> excess bandwidth factor # Use a gold code, say CA 4 d = 10000*(2*camat[4-1,:]- 1) # level shift to create 1000 * a +/-1 sequence # Create array to hold filtered output signal values x = zeros(4*len(d)) # Create array to hold FIR filter states, one less than the number of taps # CMSIS-DSP manages this for you filt_states = zeros(len(b_src)-1) CA_idx = 0 CA_code_period = 1023 for k in range(4*len(d)): if mod(k,4) == 0: # Filter one signal sample x_filt, filt_states = signal.lfilter(b_src,1,[d[ca_idx]],zi=filt_states) x[k] = x_filt else: x_filt, filt_states = signal.lfilter(b_src,1,[0],zi=filt_states) x[k] = x_filt # Increment index on CA code modulo 1023 CA_idx = mod(ca_idx+1,ca_code_period) In [72]: figure(figsize=(6,5)) subplot(211) plot(x[1000:1500]) xlabel(r Samples ) ylabel(r Amplitude Values ) 9

10 Amplitude Values title(r Filter Output Samples Upsampled and Scaled for D/A ) grid(); subplot(212) psd(x,2**10,48) # Actual fs = 48 khz ylabel(r PSD (db/hz) ) xlabel(r Frequency (khz) ) tight_layout() Filter Output Samples Upsampled and Scaled for D/A Samples Frequency (khz) PSD (db/hz) Support Code and Examples Function that Writes Gold Code Headers In [6]: def CA_code_header(fname_out,Nca): """ Write 1023 bit CA (Gold) Code Header Files Mark Wickert February 2015 """ ca = loadtxt( ca1thru37.txt,dtype=int16,usecols=(nca-1,),unpack=true) M = 1023 # code period N = 23 # code bits per line 10

11 Sca = ca + str(nca) f = open(fname_out, wt ) f.write( //define a CA code\n\n ) f.write( #include <stdint.h>\n\n ) f.write( #ifndef N_CA\n ) f.write( #define N_CA %d\n % M) f.write( #endif\n ) f.write( /*******************************************************************/\n ); f.write( /* 1023 Bit CA Gold Code %2d */\n \ % Nca); f.write( int8_t ca%d[n_ca] = { % Nca) kk = 0; for k in range(m): #k_mod = k % M if (kk < N-1) and (k < M-1): f.write( %d, % ca[k]) kk += 1 elif (kk == N-1) & (k < M-1): f.write( %d,\n % ca[k]) if k < M: if Nca < 10: f.write( ) else: f.write( ) kk = 0 else: f.write( %d % ca[k]) f.write( };\n ) f.write( /*******************************************************************/\n ) f.close() In [ ]: CA_code_header( CA_1.h,1) #write a header for CA code 1 CA_code_header( CA_12.h,12) #write a header for CA code 12 In [6]: import digitalcom as dc A fundamental property of both M-sequences and Gold codes is that they exhibit a strong correlation peak once per code period. For discrete-time signals the cross-correlation takes the form R ij [k] = 1 N N x i [n]x j [n + k] (3) n=0 where N is the data record length used in the calculation (actually estimation). Since the Gold codes form a family of codes, taking any pair codes i j with result in only a small cross-correlation value. This means that i j codes are nearly orthogonal and as signals can lie on top of each other cause minimal interference when a receiver uses the code of interest to recover via crosscorrelation the information riding the transmitted signal. Here we use the function Rij, lags axis = dc.xcorr(xi,xj,lag value range) to calculate the auto and cross-correlation between CA codes 1 and 2. The code module digitalcom.py contains the needed function. In [7]: R11,lags = dc.xcorr(2*camat[0,:]-1,2*camat[0,:]-1,100) R12,lags = dc.xcorr(2*camat[0,:]-1,2*camat[1,:]-1,100) In [73]: plot(lags,r11.real) plot(lags,r12.real) 11

12 xlabel(r Lag $k$ in auto/coss-correlation ) ylabel(r Normalized Correlation Amplitude ) title(r CA Code Correlation Properties Over One Period ) legend((r Autocorr of CA1,r Crosscorr of CA1/2 ),loc= best,) grid(); Normalized Correlation Amplitude CA Code Correlation Properties Over One Period Autocorr of CA1 Crosscorr of CA1/ Lag k in auto/coss-correlation Write FIR Header Files In [11]: def FIR_header(fname_out,h): """ Write FIR Filter Header Files Mark Wickert February 2015 """ M = len(h) N = 3 # Coefficients per line f = open(fname_out, wt ) f.write( //define a FIR coeffient Array\n\n ) f.write( #include <stdint.h>\n\n ) f.write( #ifndef M_FIR\n ) f.write( #define M_FIR %d\n % M) f.write( #endif\n ) f.write( /************************************************************************/\n ); f.write( /* FIR Filter Coefficients */\n ); f.write( float32_t h_fir[m_fir] = { ) kk = 0; for k in range(m): 12

13 #k_mod = k % M if (kk < N-1) and (k < M-1): f.write( %15.12f, % h[k]) kk += 1 elif (kk == N-1) & (k < M-1): f.write( %15.12f,\n % h[k]) if k < M: f.write( ) kk = 0 else: f.write( %15.12f % h[k]) f.write( };\n ) f.write( /************************************************************************/\n ) f.close() Write Raised Cosine and Root Raised Cosine Headers for 4 Samples/Bit In [10]: b_src = dc.sqrt_rc_imp(4,0.35) # 0.35 <==> excess bandwidth factor b_rc = dc.rc_imp(4,0.35) # 0.35 <==> excess bandwidth factor In [74]: stem(b_rc) title(r Raised Cosine Pulse Shape (Impulse Response) ) ylabel(r Amplitude ) xlabel(r Samples ) grid(); 1.0 Raised Cosine Pulse Shape (Impulse Response) Amplitude Samples 13

14 Process Data In a CSV File Exported from Analog Discovery In [94]: t_rc,xl_rc,xr_rc = loadtxt( AD_test.csv,delimiter=,,unpack=True,skiprows=6) The first 6 rows contain file data, i.e., #Digilent WaveForms Oscilloscope Acquisition #Device Name: Discovery2 #Serial Number: SN:210321A1882E #Date Time: :16:: Time (s),channel 1 (V),Channel 2 (V) , , , , , , In this case 8000 samples were captured at a sampling rate of 80 khz. Time Domain In [104]: plot(t_rc[3200:4800]*1000,xl_rc[3200:4800]) title(r Waveform Captured From Codec Output ) ylabel(r Amplitude ) xlabel(r Time (ms) ) grid(); 0.3 Waveform Captured From Codec Output Amplitude Time (ms) 14

15 Eye Plot In [105]: # Eyeplot obtain by resampling capture at 80 khz back to 48 khz # which results in 4 samples per bit (clock errors involved here) dc.eye_plot(dc.farrow_resample(xl_rc[1000:2000],80,48),2*4) Out[105]: Eye Plot Amplitude Time Index - n Power Spectral Density Estimate Using Short 8192 Data Record In [106]: psd(xl_rc,2**10,80); xlabel(r Frequency (khz) ) Out[106]: <matplotlib.text.text at 0xbe68b38> 15

16 20 Power Spectral Density (db/hz) Frequency (khz) Get on with the Problem 4 Lab Write Up In [ ]: 16