EE123 Digital Signal Processing

Transcription

1 Announcemens EE23 Digial Signal Processing Lecure Miderms: 2/28, 3/2, 4/2 HAM: Obain FRN number from FCC Fill form applicaion for amaeur radio operaor and bring o class on Friday or before (o Frank) Ink only, prin carefully. Fill phone and and address please Secion 2 BLANK Will pos form on websie oo. Lecure on ham omorrow 6-7pm HP audiorium 2 How do you know his guy is insane? Las Time Sared wih STFT Heisenberg Boxes Coninue and move o waveles Specrum no symmeric, so ca mus be imaginary hp://xkcd.com/26/ 3 4

2 Discree Transforms (Finie) DFT is only one ou of a LARGE class of ransforms Used for: Analysis Compression Denoising Deecion Recogniion Approximaion (Sparse) Sparse represenaion has been one of hoes research opics in las years in sp Example of specral analysis Specrum of a bird chirping Ineresing,... bu... Does no ell whole sory No emporal informaion x[n] n Specrum of a bird chirp Hz x 4 6 Time Dependen Fourier Transform To ge emporal informaion, use par of signal around every ime poin Time Dependen Fourier Transform To ge emporal informaion, use par of signal around every ime poin X[n, ) = X m= x[n + m]w[m]e jm X[n, ) = X m= x[n + m]w[m]e jm *Also called Shor-ime Fourier Transform (STFT) *Also called Shor-ime Fourier Transform (STFT) Mapping from D 2D, n discree, w con. Simply slide a window and compue DTFT 7 8

3 Specrogram Discree Time Dependen FT X r [k] = LX m= L - Window lengh R - Jump of samples N - DFT lengh x[rr + m]w[m]e j2 km/n Tradeoff beween ime and frequency resoluion Heisenberg Boxes Time-Frequency uncerainy principle 2 hp:// DFT X[k] = NX n= x[n]e j2 kn/n = 2 N = N =2 one DFT coefficien 2

4 DFT = 2 N = N X[k] = NX n= x[n]e j2 kn/n Discree STFT X[r, k] = = 2 L = L LX m= opional x[rr + m]w[m]e j2 km/n =2 Quesion: Wha is effec of zero-padding? Answer: Overlapped Tiling 3 one STFT coefficien 4 Discree STFT X[r, k] = LX m= opional x[rr + m]w[m]e j2 km/n Applicaions Time Frequency Analysis Specrogram of Orca whale 4 = 2 L = L 3 2 Quesion: Wha is effec of R on iling? wha effec of N? Answer: Overlapping in ime of frequency or boh

5 Specrogram (A) (B) Wha is difference beween specrograms? a) Window size B<A c) Window ype is differen b) Window size B>A d) (A) uses overlapping window Sidelobes of Hann vs recangular window W(e j T ) 2 DTFT of Hamming Window -2-2 DTFT of Sampled, Windowed Signal W(e j T ) DTFT of Recangular Window V(e j T ) 8 Specrogram Miki Lusig UCB. Based on Course Noes by J.M Kahn Wha is difference beween specrograms? a) Window size B<A b) Window size B>A (A) (B) c) Window ype is differen d) (A) uses overlapping window Windows Examples Specrogram Fall 2, EE23 Digial Signal Processing Hamming Window, L = 32 w[n] Hamming Window, L = 32.2 Windows Examples n Hamming Window, L = 64 v[n] w[n] Sampled, Windowed Signal, Hamming Window, L = 32 Hamming Window, L = n Miki Lusig UCB. Based on Course n Noes by J.M Kahn W(e j T ) V(e j T ) W(e j T ) DTFT of Hamming Window -2-2 DTFT of Sampled, Windowed Signal DTFT of Hamming Window Fall 2, EE23 Digial Signal Processing Sampled, Windowed Signal, Hamming Window, L = DTFT of Sampled, Windowed Signal. 2 v[n] -. V(e j T )

6 will firs look a radio FM specrum. In US broadcas FM radio band is Mhz. spli ino 2KHz slos. This is relaively a large bandwidh and refor i is also called wideband as opposed o narrowband FM which can be as low as Khz. In FM radio informaion is encoded Now, SDR already demodulaes by carrier frequency fc, so samples you ge are acually ofcarrier, FM Specrogram of FM radio Baseband odulaing Specrogram frequency of Z Z nt y[n] = y(nt ) = A exp j2 f x( )d, yc () = A cos 2 fc + 2 f x( )d. hp://en.wikipedia.org/wiki/fm_broadcasing for more informaion). The baseband signal is: x() = (L + R) +. cos(2 fp ) + (L R) cos(2 (2fp )) +. RBDS() cos(2 (3fp )). w, SDR already demodulaes by carrier frequency fc, so samples you ge are acually {z } {z } {z } {z } mono sereo pilo digial RBDS where T is sampling x() = (Lrae. + R) +. cos(2 fp ) + (L R) cos(2 (2fp )) +. RBDS() cos(2 (3fp )). Z, fc is carrier frequency, f is frequency deviaion and {z The}signal RBDS(){zis a signal }consiss nt x() is a normalized baseband signal. of ± a consan {z inervals which } encode, and bis. This {z is } mono specrum of x(): pilo sereo digial RBDS y[n] = y(nt ) = A exp j2 f x( )d, Pick your favorie radio saion (make sure re is good recepion where you live Don ry i in broadcas FM baseband signal, x(), consiss of mono (Lef+Righ) Channels from 3Hz o KHz, The signal RBDS() is a signal consiss of ± consan inervals which good encoderecepion, and bis. is basemen or in rooms inside Cory ha do no havea windows). I ve esed fromthis Cory hall o signal a fsampling ampliude modulaed Sereo (Lef - Righ) channels around 2 fp = 38KHz specrum of x(): p = KHz, ere T is rae. h floor a 88.Mhz and 4.Mhz. Acquire 4 seconds of daa a a sampling rae of 2, Hz. You Specrogram of FM radio digial RBDS, which encodes saion informaion, song name ec. a 3 fp = 7KHz. (See Compue can use eir rl_cp or rl_sdr for ha. and display a specrogram wih a window of 88.6 k your favorie radio saion (make sure re is good recepion where you live Don ry isamples. in Include specrogram in your repor. Wha is specral and emporal resoluion? Explain emen or in rooms inside Cory ha do no have windows). I ve esed good recepion from Cory whahall you see. Don forge o play wih di eren dynamic ranges of specrogram for bes visualizaion. floor a 88.Mhz and 4.Mhz. Acquire 4 seconds of daa a a sampling rae of 2, Hz. You 8 Specrogram ofalready Demodulaed FMfrequency radio (Adele 6. MHz) use eir rl_cp or rl_sdr for ha. Compue and display a specrogram wih a window of Now, SDR demodulaes by carrier fc, so sampleson you ge are acually Z nt mples. Include specrogram in your repor. Wha is specral and emporal resoluion? In Explain many analog radio sysems, FM demodulaion is performedx( by implemening a phased-locked loop 88. y[n] = y(nt ) = A exp j2 f )d, 7KHz a you see. Don forge o play wih di eren dynamic ranges of specrogram for bes visualizaion. ha racks frequency of signal and exrac message x(). However, i is very easy o implemen hp://en.wikipedia.org/wiki/fm_broadcasing for more informaion). The baseband signal is: where T is sampling rae. an FM demodulaion when samples are discree. Noe, ha signal x() is proporional o 38KHz Pick your favorie radio saion (make sure re is good recepion where you live Don ry i in insananeous frequency of signal. I is also proporional o derivaive of phase We can implebasemen or in rooms inside Cory ha do no have windows). I ve esed good recepion from Cory hall many analog radio sysems, FM demodulaion is performed by implemening a phased-locked loop floor a 88.Mhz and 4.Mhz. Acquire 4 seconds daa signal a a sampling rae of 2, Hz. You men his by compuing h finie di erences of phase of ofour angle(y[n])-angle(y[n-]). An KHz can use eir rl_cp or rl_sdr for ha. Compue and display a specrogram wih a window of racks frequency of signal and exrac message x(). However, i is very easy o implemen [n samples. Include specrogram in your repor. Wha is produc specral and emporal resoluion? alernaive ha is robus o phase wrapping is o compue y[n]y ]Explain and n compue is 88.4 wha you see. Don forge o play wih di eren dynamic ranges of specrogram for bes visualizaion. FM demodulaion o Convince yourself = when samples are discree. Noe, ha signal x() is proporional =2sec phase. ha demodulaes i works In by malab i can be simply implemened by: ge are acually Now, SDR already carrier frequency fc, so samples you ananeous frequency of signal. I is also proporional o derivaive of M. phase We can implelusig, EECS UC Berkeley M. Lusig, In many analog radio sysems, FM demodulaion is performedzby implemening a phased-locked loop EECS UC Berkeley >>2 xan = angle(y[2:end].*conj(y[:end-])); n his by compuing finie di erences of phase of our signal angle(y[n])-angle(y[n-]). ha racks frequency of signal and exrac message x(). nt However, i is very easy o implemen 22 y[n] = y(nt ) = A exp j2 f x( )d, an FM demodulaion when samples are discree. Noe, ha signal x() is proporional o rnaive ha is robus o phase wrapping is o compue produc y[n]y [n ] and n compue is insananeous frequency of signal. I is also proporional o derivaive of phase We can implefm demodulae samples you from SDR display specrogram. Noe, ha aferfm men his bygo compuing finie di erences ofand phase of our signal angle(y[n])-angle(y[n-]). An se. Convince yourself ha i works In malab i can be simply implemened by: alernaive ha is robus o phase wrapping is o compue produc y[n]y [n ] and n compue is demodulaing should be real and hence only half specrum should be displayed. Idenify where T issignal sampling phase. rae. Convince yourself ha i works In malab i can be simply implemened by: mono audio, pilo, sereo and RBDS signals. Noe, ha RBDS signal may be oo weak o x = angle(y[2:end].*conj(y[:end-])); >> x = angle(y[2:end].*conj(y[:end-])); Pick Applicaions your radio saion sure re is good recepion where you deec or may need favorie beer specral resoluion. FM demodulae (make samples you go from SDR and display specrogram. Noe, ha live aferfmdon ry i in Subcarrier FM radio (Hidden Radio Saions) signalha shoulddo be real andhave hence only half specrum should be displayed. demodulae samples you go from SDR and display specrogram. Noe, ha aferfm basemen or in roomsdemodulaing inside Cory no windows). I ve esed goodidenify recepion from Cory hall mono audio, pilo, sereo and RBDS signals. Noe, ha RBDS signal may be oo weak o modulaing signal should be real and hence only half specrum should be displayed. Idenify h floor a 88.Mhzdeec andor may 4.Mhz. Acquire 4 seconds of daa a a sampling rae of 2, Hz. You need beer specral resoluion. and display a specrogram wih a window of no audio, pilo, sereo and RBDS signals. Noe, ha RBDS signal may be oo weak o can use eir rl_cp or rl_sdr for ha. Compue samples. Include specrogram in your repor. Wha is specral and emporal resoluion? Explain ec or may need beer specral resoluion. Time Frequency Analysis wha you see. Don forge o play wih di eren dynamic ranges of specrogram for bes visualizaion. In many analog radio sysems, FM demodulaion is performed by implemening a phased-locked loop ha racksspecrogram frequency ofof signalcommunicaions and exrac message x(). However, is very easy o implemen digial - Frequency ShifiKeying an FM demodulaion when samples are discree. Noe, ha signal x() is proporional o insananeous frequency of signal. I is also proporional o derivaive of phase We can implemen his by compuing finie di erences of phase of our signal angle(y[n])-angle(y[n-]). An alernaive ha is robus o phase wrapping is o compue produc y[n]y [n ] and n compue is phase. Convince yourself ha i works In malab i can be simply implemened by: >> x = angle(y[2:end].*conj(y[:end-])); FM demodulae samples you go from SDR and display specrogram. Noe, ha aferfm demodulaing signal should be real and hence only half specrum should be displayed. Idenify mono audio, pilo, sereo and RBDS signals. Noe, ha RBDS signal may be oo weak o deec or= may need beer specral resoluion. =sec 23 24

7 STFT Reconsrucion x[rr + m]w L [m] = N NX k= X[n, k]e j2 km/n For non-overlapping windows, R=L : x[n] = Wha is problem? x[n rl] w L [n rl] rl apple n apple (r + )R STFT Reconsrucion x[rr + m]w L [m] = N NX k= X[n, k]e j2 km/n For non-overlapping windows, R=L : x[n] = x[n rl] w L [n rl] rl apple n apple (r + )R For sable reconsrucion mus overlap window % (a leas) 2 26 STFT Reconsrucion Applicaions For sable reconsrucion mus overlap window % (a leas) Noise removal For Hann, Barle reconsruc wih overlap and add. No division Recall bird chirp x[n] n 6 Specrum of a bird chirp Hz x

8 Applicaion Denoising of Sparse specrograms Limiaions of Discree STFT Need overlapping No orhogonal Compuaionally inensive O(MN log N) Same size Heisenberg boxes Specrum is sparse can implemen adapive filer, or jus hreshold 2 3 From STFT o Waveles Basic Idea: low-freq changes slowly - fas racking unimporan Fas racking of high-freq is imporan in many apps. Mus adap Heisenberg box o frequency Back o coninuous ime for a bi... 3