(1) (2) DSP and Digital Filters (2015-7310) FM Radio: 14 1 / 12
FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters (2015-7310) FM Radio: 14 2 / 12
FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz Each channel: ±100 khz [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters (2015-7310) FM Radio: 14 2 / 12
FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz Each channel: ±100 khz Baseband signal: Mono (L + R):±15kHz [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters (2015-7310) FM Radio: 14 2 / 12
FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz Each channel: ±100 khz Baseband signal: Mono (L + R):±15kHz Stereo (L R):38±15kHz [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters (2015-7310) FM Radio: 14 2 / 12
FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz Each channel: ±100 khz Baseband signal: Mono (L + R):±15kHz Pilot tone: 19kHz Stereo (L R):38±15kHz [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters (2015-7310) FM Radio: 14 2 / 12
FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz Each channel: ±100 khz Baseband signal: Mono (L + R):±15kHz Pilot tone: 19kHz Stereo (L R):38±15kHz RDS:57±2kHz [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters (2015-7310) FM Radio: 14 2 / 12
FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz Each channel: ±100 khz Baseband signal: Mono (L + R):±15kHz Pilot tone: 19kHz Stereo (L R):38±15kHz RDS:57±2kHz FM Modulation: [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters (2015-7310) FM Radio: 14 2 / 12
FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz Each channel: ±100 khz Baseband signal: Mono (L + R):±15kHz Pilot tone: 19kHz Stereo (L R):38±15kHz RDS:57±2kHz FM Modulation: Freq deviation: ±75 khz [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters (2015-7310) FM Radio: 14 2 / 12
FM Radio Block Diagram (1) (2) FM spectrum: 87.5 to108mhz Each channel: ±100 khz Baseband signal: Mono (L + R):±15kHz Pilot tone: 19kHz Stereo (L R):38±15kHz RDS:57±2kHz FM Modulation: Freq deviation: ±75 khz L R signal is multiplied by38khz to shift it to baseband [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters (2015-7310) FM Radio: 14 2 / 12
Aliased ADC (1) (2) FM band: 87.5 to108mhz Normally sample atf s > 2f DSP and Digital Filters (2015-7310) FM Radio: 14 3 / 12
Aliased ADC (1) (2) FM band: 87.5 to108mhz Normally sample atf s > 2f However: DSP and Digital Filters (2015-7310) FM Radio: 14 3 / 12
Aliased ADC (1) (2) FM band: 87.5 to108mhz Normally sample atf s > 2f However: f s = 80MHz aliases band down to[7.5, 28]MHz. DSP and Digital Filters (2015-7310) FM Radio: 14 3 / 12
Aliased ADC (1) (2) FM band: 87.5 to108mhz Normally sample atf s > 2f However: f s = 80MHz aliases band down to[7.5, 28]MHz. ve frequencies alias to[ 28, 7.5] MHz. DSP and Digital Filters (2015-7310) FM Radio: 14 3 / 12
Aliased ADC (1) (2) FM band: 87.5 to108mhz Normally sample atf s > 2f However: f s = 80MHz aliases band down to[7.5, 28]MHz. ve frequencies alias to[ 28, 7.5] MHz. We must suppress other frequencies that alias to the range ±[7.5, 28] MHz. DSP and Digital Filters (2015-7310) FM Radio: 14 3 / 12
Aliased ADC (1) (2) FM band: 87.5 to108mhz Normally sample atf s > 2f However: f s = 80MHz aliases band down to[7.5, 28]MHz. ve frequencies alias to[ 28, 7.5] MHz. We must suppress other frequencies that alias to the range ±[7.5, 28] MHz. Need an analogue bandpass filter to extract the FM band. DSP and Digital Filters (2015-7310) FM Radio: 14 3 / 12
Aliased ADC (1) (2) FM band: 87.5 to108mhz Normally sample atf s > 2f However: f s = 80MHz aliases band down to[7.5, 28]MHz. ve frequencies alias to[ 28, 7.5] MHz. We must suppress other frequencies that alias to the range ±[7.5, 28] MHz. Need an analogue bandpass filter to extract the FM band. Transition band mid-points are atf s = 80MHz and1.5f s = 120MHz. DSP and Digital Filters (2015-7310) FM Radio: 14 3 / 12
Aliased ADC (1) (2) FM band: 87.5 to108mhz Normally sample atf s > 2f However: f s = 80MHz aliases band down to[7.5, 28]MHz. ve frequencies alias to[ 28, 7.5] MHz. We must suppress other frequencies that alias to the range ±[7.5, 28] MHz. Need an analogue bandpass filter to extract the FM band. Transition band mid-points are atf s = 80MHz and1.5f s = 120MHz. You can use an aliased analog-digital converter (ADC) provided that the target band fits entirely between two consecutive multiples of 1 2 f s. DSP and Digital Filters (2015-7310) FM Radio: 14 3 / 12
Aliased ADC (1) (2) FM band: 87.5 to108mhz Normally sample atf s > 2f However: f s = 80MHz aliases band down to[7.5, 28]MHz. ve frequencies alias to[ 28, 7.5] MHz. We must suppress other frequencies that alias to the range ±[7.5, 28] MHz. Need an analogue bandpass filter to extract the FM band. Transition band mid-points are atf s = 80MHz and1.5f s = 120MHz. You can use an aliased analog-digital converter (ADC) provided that the target band fits entirely between two consecutive multiples of 1 2 f s. Lower ADC sample rate. Image = undistorted frequency-shifted copy. DSP and Digital Filters (2015-7310) FM Radio: 14 3 / 12
Channel Selection (1) (2) FM band shifted to7.5 to28mhz (from87.5 to108mhz) DSP and Digital Filters (2015-7310) FM Radio: 14 4 / 12
Channel Selection (1) (2) FM band shifted to7.5 to28mhz (from87.5 to108mhz) We need to select a single channel 200kHz wide DSP and Digital Filters (2015-7310) FM Radio: 14 4 / 12
Channel Selection (1) (2) FM band shifted to7.5 to28mhz (from87.5 to108mhz) We need to select a single channel 200kHz wide We shift selected channel to DC and then downsample tof s = 400kHz. Assume channel centre frequency isf c = c 100kHz DSP and Digital Filters (2015-7310) FM Radio: 14 4 / 12
Channel Selection (1) (2) FM band shifted to7.5 to28mhz (from87.5 to108mhz) We need to select a single channel 200kHz wide We shift selected channel to DC and then downsample tof s = 400kHz. Assume channel centre frequency isf c = c 100kHz We must apply a filter before downsampling to remove unwanted images DSP and Digital Filters (2015-7310) FM Radio: 14 4 / 12
Channel Selection (1) (2) FM band shifted to7.5 to28mhz (from87.5 to108mhz) We need to select a single channel 200kHz wide We shift selected channel to DC and then downsample tof s = 400kHz. Assume channel centre frequency isf c = c 100kHz We must apply a filter before downsampling to remove unwanted images The downsampled signal is complex since positive and negative frequencies contain different information. DSP and Digital Filters (2015-7310) FM Radio: 14 4 / 12
Channel Selection (1) (2) FM band shifted to7.5 to28mhz (from87.5 to108mhz) We need to select a single channel 200kHz wide We shift selected channel to DC and then downsample tof s = 400kHz. Assume channel centre frequency isf c = c 100kHz We must apply a filter before downsampling to remove unwanted images The downsampled signal is complex since positive and negative frequencies contain different information. We will look at three methods: 1 Freq shift, then polyphase lowpass filter DSP and Digital Filters (2015-7310) FM Radio: 14 4 / 12
Channel Selection (1) (2) FM band shifted to7.5 to28mhz (from87.5 to108mhz) We need to select a single channel 200kHz wide We shift selected channel to DC and then downsample tof s = 400kHz. Assume channel centre frequency isf c = c 100kHz We must apply a filter before downsampling to remove unwanted images The downsampled signal is complex since positive and negative frequencies contain different information. We will look at three methods: 1 Freq shift, then polyphase lowpass filter 2 Polyphase bandpass complex filter DSP and Digital Filters (2015-7310) FM Radio: 14 4 / 12
Channel Selection (1) (2) FM band shifted to7.5 to28mhz (from87.5 to108mhz) We need to select a single channel 200kHz wide We shift selected channel to DC and then downsample tof s = 400kHz. Assume channel centre frequency isf c = c 100kHz We must apply a filter before downsampling to remove unwanted images The downsampled signal is complex since positive and negative frequencies contain different information. We will look at three methods: 1 Freq shift, then polyphase lowpass filter 2 Polyphase bandpass complex filter 3 Polyphase bandpass real filter DSP and Digital Filters (2015-7310) FM Radio: 14 4 / 12
Channel Selection (1) (1) (2) Multiply bye j2πr fc 80MHz to shift channel atf c to DC. f c = c 100k f c 80M = c 800 DSP and Digital Filters (2015-7310) FM Radio: 14 5 / 12
Channel Selection (1) (1) (2) Multiply bye j2πr fc 80MHz to shift channel atf c to DC. f c = c 100k f c 80M = c 800 Result of multiplication is complex (thick lines on diagram) DSP and Digital Filters (2015-7310) FM Radio: 14 5 / 12
Channel Selection (1) (1) (2) Multiply bye j2πr fc 80MHz to shift channel atf c to DC. f c = c 100k f c 80M = c 800 Result of multiplication is complex (thick lines on diagram) Next, lowpass filter to ±100 khz ω = 2π 200 k 80 M = 0.157 DSP and Digital Filters (2015-7310) FM Radio: 14 5 / 12
Channel Selection (1) (1) (2) Multiply bye j2πr fc 80MHz to shift channel atf c to DC. f c = c 100k f c 80M = c 800 Result of multiplication is complex (thick lines on diagram) Next, lowpass filter to ±100 khz ω = 2π 200 k 80 M = 0.157 M = 60 db 3.5 ω = 1091 DSP and Digital Filters (2015-7310) FM Radio: 14 5 / 12
Channel Selection (1) (1) (2) Multiply bye j2πr fc 80MHz to shift channel atf c to DC. f c = c 100k f c 80M = c 800 Result of multiplication is complex (thick lines on diagram) Next, lowpass filter to ±100 khz ω = 2π 200 k 80 M = 0.157 M = 60 db 3.5 ω = 1091 Finally, downsample 200 : 1 DSP and Digital Filters (2015-7310) FM Radio: 14 5 / 12
Channel Selection (1) (1) (2) Multiply bye j2πr fc 80MHz to shift channel atf c to DC. f c = c 100k f c 80M = c 800 Result of multiplication is complex (thick lines on diagram) Next, lowpass filter to ±100 khz ω = 2π 200 k 80 M = 0.157 M = 60 db 3.5 ω = 1091 Finally, downsample 200 : 1 Polyphase: H p (z) 1092 has 200 = 6 taps DSP and Digital Filters (2015-7310) FM Radio: 14 5 / 12
Channel Selection (1) (1) (2) Multiply bye j2πr fc 80MHz to shift channel atf c to DC. f c = c 100k f c 80M = c 800 Result of multiplication is complex (thick lines on diagram) Next, lowpass filter to ±100 khz ω = 2π 200 k 80 M = 0.157 M = 60 db 3.5 ω = 1091 Finally, downsample 200 : 1 Polyphase: H p (z) 1092 has 200 = 6 taps Complex data Real Coefficients (needs 2 multiplies per tap) DSP and Digital Filters (2015-7310) FM Radio: 14 5 / 12
Channel Selection (1) (1) (2) Multiply bye j2πr fc 80MHz to shift channel atf c to DC. f c = c 100k f c 80M = c 800 Result of multiplication is complex (thick lines on diagram) Next, lowpass filter to ±100 khz ω = 2π 200 k 80 M = 0.157 M = 60 db 3.5 ω = 1091 Finally, downsample 200 : 1 Polyphase: H p (z) 1092 has 200 = 6 taps Complex data Real Coefficients (needs 2 multiplies per tap) Multiplication Load: 2 80MHz (freq shift) +12 80MHz (H p (z)) =14 80MHz DSP and Digital Filters (2015-7310) FM Radio: 14 5 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 We multiplyu[r] bye j2πr c 800, convolve withh[m] and then downsample: v[n] = M m=0 h[m]u[200n m]e j2π(200n m) c 800 [r = 200n] DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 We multiplyu[r] bye j2πr c 800, convolve withh[m] and then downsample: v[n] = M m=0 h[m]u[200n m]e j2π(200n m) c 800 [r = 200n] = M mc m=0h[m]ej2π 800u[200n m]e j2π200n4k+l 800 [c = 4k +1] DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 We multiplyu[r] bye j2πr c 800, convolve withh[m] and then downsample: v[n] = M m=0 h[m]u[200n m]e j2π(200n m) c 800 [r = 200n] = M mc m=0h[m]ej2π 800u[200n m]e j2π200n4k+l 800 [c = 4k +1] = M m=0 g [c][m]u[200n m]e j2π ln 4 [g [c] [m] = mc j2π h[m]e 800 ] DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 We multiplyu[r] bye j2πr c 800, convolve withh[m] and then downsample: v[n] = M m=0 h[m]u[200n m]e j2π(200n m) c 800 [r = 200n] = M mc m=0h[m]ej2π 800u[200n m]e j2π200n4k+l 800 [c = 4k +1] = M m=0 g [c][m]u[200n m]e j2π ln 4 [g [c] [m] = mc j2π h[m]e 800 ] = ( j) ln M m=0 g [c][m]u[200n m] [e j2π ln 4 indep ofm] DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 We multiplyu[r] bye j2πr c 800, convolve withh[m] and then downsample: v[n] = M m=0 h[m]u[200n m]e j2π(200n m) c 800 [r = 200n] = M mc m=0h[m]ej2π 800u[200n m]e j2π200n4k+l 800 [c = 4k +1] = M m=0 g [c][m]u[200n m]e j2π ln 4 [g [c] [m] = mc j2π h[m]e 800 ] = ( j) ln M m=0 g [c][m]u[200n m] [e j2π ln 4 indep ofm] DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 We multiplyu[r] bye j2πr c 800, convolve withh[m] and then downsample: v[n] = M m=0 h[m]u[200n m]e j2π(200n m) c 800 [r = 200n] = M mc m=0h[m]ej2π 800u[200n m]e j2π200n4k+l 800 [c = 4k +1] = M m=0 g [c][m]u[200n m]e j2π ln 4 [g [c] [m] = mc j2π h[m]e 800 ] = ( j) ln M m=0 g [c][m]u[200n m] [e j2π ln 4 indep ofm] Multiplication Load for polyphase implementation: G [c],p (z) has complex coefficients real input 2mults per tap DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 We multiplyu[r] bye j2πr c 800, convolve withh[m] and then downsample: v[n] = M m=0 h[m]u[200n m]e j2π(200n m) c 800 [r = 200n] = M mc m=0h[m]ej2π 800u[200n m]e j2π200n4k+l 800 [c = 4k +1] = M m=0 g [c][m]u[200n m]e j2π ln 4 [g [c] [m] = mc j2π h[m]e 800 ] = ( j) ln M m=0 g [c][m]u[200n m] [e j2π ln 4 indep ofm] Multiplication Load for polyphase implementation: G [c],p (z) has complex coefficients real input 2mults per tap ( j) ln {+1, j, 1, +j} so no actual multiplies needed DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (2) (1) (2) Channel centre frequency f c = c 100kHz wherecis an integer. Writec = 4k +l wherek = c 4 andl = cmod 4 We multiplyu[r] bye j2πr c 800, convolve withh[m] and then downsample: v[n] = M m=0 h[m]u[200n m]e j2π(200n m) c 800 [r = 200n] = M mc m=0h[m]ej2π 800u[200n m]e j2π200n4k+l 800 [c = 4k +1] = M m=0 g [c][m]u[200n m]e j2π ln 4 [g [c] [m] = mc j2π h[m]e 800 ] = ( j) ln M m=0 g [c][m]u[200n m] [e j2π ln 4 indep ofm] Multiplication Load for polyphase implementation: G [c],p (z) has complex coefficients real input 2mults per tap ( j) ln {+1, j, 1, +j} so no actual multiplies needed Total: 12 80MHz (forg [c],p (z)) +0(for j ln ) =12 80MHz DSP and Digital Filters (2015-7310) FM Radio: 14 6 / 12
Channel Selection (3) (1) (2) Channel frequencyf c = c 100kHz wherec = 4k +l is an integer DSP and Digital Filters (2015-7310) FM Radio: 14 7 / 12
Channel Selection (3) (1) (2) Channel frequencyf c = c 100kHz wherec = 4k +l is an integer cm j2π g [c] [m] = h[m]e 800 DSP and Digital Filters (2015-7310) FM Radio: 14 7 / 12
Channel Selection (3) (1) (2) Channel frequencyf c = c 100kHz wherec = 4k +l is an integer cm j2π g [c] [m] = h[m]e 800 c(200s+p) j2π g [c],p [s] = g c [200s+p]= h[200s+p]e 800 [polyphase] DSP and Digital Filters (2015-7310) FM Radio: 14 7 / 12
Channel Selection (3) (1) (2) Channel frequencyf c = c 100kHz wherec = 4k +l is an integer cm j2π g [c] [m] = h[m]e 800 c(200s+p) j2π g [c],p [s] = g c [200s+p]= h[200s+p]e 800 [polyphase] = h[200s+p]e j2π cs cp 4 j2π e 800 DSP and Digital Filters (2015-7310) FM Radio: 14 7 / 12
Channel Selection (3) (1) (2) Channel frequencyf c = c 100kHz wherec = 4k +l is an integer cm j2π g [c] [m] = h[m]e 800 c(200s+p) j2π g [c],p [s] = g c [200s+p]= h[200s+p]e 800 [polyphase] = h[200s+p]e j2π cs cp 4 j2π e 800 h[200s+p]e j2π cs 4 α p DSP and Digital Filters (2015-7310) FM Radio: 14 7 / 12
Channel Selection (3) (1) (2) Channel frequencyf c = c 100kHz wherec = 4k +l is an integer cm j2π g [c] [m] = h[m]e 800 c(200s+p) j2π g [c],p [s] = g c [200s+p]= h[200s+p]e 800 [polyphase] = h[200s+p]e j2π cs cp 4 j2π e 800 h[200s+p]e j2π cs 4 α p (4k+l)s j2π Definef [c],p [s] = h[200s+p]e 4 = j ls h[200s+p] DSP and Digital Filters (2015-7310) FM Radio: 14 7 / 12
Channel Selection (3) (1) (2) Channel frequencyf c = c 100kHz wherec = 4k +l is an integer cm j2π g [c] [m] = h[m]e 800 c(200s+p) j2π g [c],p [s] = g c [200s+p]= h[200s+p]e 800 [polyphase] = h[200s+p]e j2π cs cp 4 j2π e 800 h[200s+p]e j2π cs 4 α p (4k+l)s j2π Definef [c],p [s] = h[200s+p]e 4 = j ls h[200s+p] Althoughf [c],p [s] is complex it requires only one multiplication per tap because each tap is either purely real or purely imaginary. DSP and Digital Filters (2015-7310) FM Radio: 14 7 / 12
Channel Selection (3) (1) (2) Channel frequencyf c = c 100kHz wherec = 4k +l is an integer cm j2π g [c] [m] = h[m]e 800 c(200s+p) j2π g [c],p [s] = g c [200s+p]= h[200s+p]e 800 [polyphase] = h[200s+p]e j2π cs cp 4 j2π e 800 h[200s+p]e j2π cs 4 α p (4k+l)s j2π Definef [c],p [s] = h[200s+p]e 4 = j ls h[200s+p] Althoughf [c],p [s] is complex it requires only one multiplication per tap because each tap is either purely real or purely imaginary. Multiplication Load: cp j2π 6 80 MHz (F p (z)) +4 80MHz ( e 800 ) =10 80MHz DSP and Digital Filters (2015-7310) FM Radio: 14 7 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ The instantaneous frequency ofv(t) is dφ dt. DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ The instantaneous frequency ofv(t) is dφ dt. We need to calculatex(t) = dφ dt = di(logv) dt DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ The instantaneous frequency ofv(t) is dφ dt. We need to calculatex(t) = dφ dt = di(logv) dt = I ( 1 v ) dv dt DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ The instantaneous frequency ofv(t) is dφ dt. We need to calculatex(t) = dφ dt = di(logv) dt = I ( 1 v ) dv dt = 1 v I ( ) v dv 2 dt DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ The instantaneous frequency ofv(t) is dφ dt. We need to calculatex(t) = dφ dt = di(logv) dt = I ( 1 v ) dv dt = 1 v I ( ) v dv 2 dt DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ The instantaneous frequency ofv(t) is dφ dt. We need to calculatex(t) = dφ dt = di(logv) dt = I ( 1 v ) dv dt = 1 v I ( ) v dv 2 dt We need: (1) Differentiation filter, D(z) DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ The instantaneous frequency ofv(t) is dφ dt. We need to calculatex(t) = dφ dt = di(logv) dt = I ( 1 v ) dv dt = 1 v I ( ) v dv 2 dt We need: (1) Differentiation filter, D(z) (2) Complex multiply,w[n] v [n] (only needipart) DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ The instantaneous frequency ofv(t) is dφ dt. We need to calculatex(t) = dφ dt = di(logv) dt = I ( 1 v ) dv dt = 1 v I ( ) v dv 2 dt We need: (1) Differentiation filter, D(z) (2) Complex multiply,w[n] v [n] (only needipart) (3) Real Divide by v 2 DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
FM Demodulator (1) (2) Complex FM signal centred at DC:v(t) = v(t) e jφ(t) We know thatlogv = log v +jφ The instantaneous frequency ofv(t) is dφ dt. We need to calculatex(t) = dφ dt = di(logv) dt = I ( 1 v ) dv dt = 1 v I ( ) v dv 2 dt We need: (1) Differentiation filter, D(z) (2) Complex multiply,w[n] v [n] (only needipart) (3) Real Divide by v 2 x[n] is baseband signal (real): DSP and Digital Filters (2015-7310) FM Radio: 14 8 / 12
Differentiation Filter (1) (2) DSP and Digital Filters (2015-7310) FM Radio: 14 9 / 12
Differentiation Filter (1) (2) Window design method: (1) calculate d[n] for the ideal filter (2) multiply by a window to give finite support DSP and Digital Filters (2015-7310) FM Radio: 14 9 / 12
Differentiation Filter (1) (2) Window design method: (1) calculate d[n] for the ideal filter (2) multiply by a window to give finite support Differentiation: d dt ejωt = jωe jωt DSP and Digital Filters (2015-7310) FM Radio: 14 9 / 12
Differentiation Filter (1) (2) Window design method: (1) calculate d[n] for the ideal filter (2) multiply by a window to give finite support Differentiation: d dt ejωt = jωe jωt D(e jω ) = { jω ω ω 0 0 ω > ω 0 DSP and Digital Filters (2015-7310) FM Radio: 14 9 / 12
Differentiation Filter (1) (2) Window design method: (1) calculate d[n] for the ideal filter (2) multiply by a window to give finite support Differentiation: Henced[n] = 1 2π d dt ejωt = jωe jωt D(e jω ) = ω0 ω 0 jωe jωn dω = j 2π [ ωe jnω jn ejnω j 2 n 2 ] ω0 = nω 0 cosnω 0 sinnω 0 πn 2 { jω ω ω 0 0 ω > ω 0 ω 0 [IDTFT] DSP and Digital Filters (2015-7310) FM Radio: 14 9 / 12
Differentiation Filter (1) (2) Window design method: (1) calculate d[n] for the ideal filter (2) multiply by a window to give finite support Differentiation: Henced[n] = 1 2π d dt ejωt = jωe jωt D(e jω ) = ω0 ω 0 jωe jωn dω = j 2π [ ωe jnω jn ejnω j 2 n 2 ] ω0 = nω 0 cosnω 0 sinnω 0 πn 2 { jω ω ω 0 0 ω > ω 0 ω 0 [IDTFT] 1.5 ω 0 0 H 1 0.5 H (db) -20-40 -60 ω 0 0 0 0.5 1 1.5 2 2.5 3 ω (rad/sample) -80 0 0.5 1 1.5 2 2.5 3 ω (rad/sample) UsingM = 18, Kaiser window,β = 7 andω 0 = 2.2 = 2π 140kHz 400 khz : DSP and Digital Filters (2015-7310) FM Radio: 14 9 / 12
Differentiation Filter (1) (2) Window design method: (1) calculate d[n] for the ideal filter (2) multiply by a window to give finite support Differentiation: Henced[n] = 1 2π d dt ejωt = jωe jωt D(e jω ) = ω0 ω 0 jωe jωn dω = j 2π [ ωe jnω jn ejnω j 2 n 2 ] ω0 = nω 0 cosnω 0 sinnω 0 πn 2 { jω ω ω 0 0 ω > ω 0 ω 0 [IDTFT] 1.5 ω 0 0 H 1 0.5 H (db) -20-40 -60 ω 0 0 0 0.5 1 1.5 2 2.5 3 ω (rad/sample) -80 0 0.5 1 1.5 2 2.5 3 ω (rad/sample) UsingM = 18, Kaiser window,β = 7 andω 0 = 2.2 = 2π 140kHz 400 khz : Near perfect differentiation forω 1.6 ( 100kHz forf s = 400kHz) DSP and Digital Filters (2015-7310) FM Radio: 14 9 / 12
Differentiation Filter (1) (2) Window design method: (1) calculate d[n] for the ideal filter (2) multiply by a window to give finite support Differentiation: Henced[n] = 1 2π d dt ejωt = jωe jωt D(e jω ) = ω0 ω 0 jωe jωn dω = j 2π [ ωe jnω jn ejnω j 2 n 2 ] ω0 = nω 0 cosnω 0 sinnω 0 πn 2 { jω ω ω 0 0 ω > ω 0 ω 0 [IDTFT] 1.5 ω 0 0 H 1 0.5 H (db) -20-40 -60 ω 0 0 0 0.5 1 1.5 2 2.5 3 ω (rad/sample) -80 0 0.5 1 1.5 2 2.5 3 ω (rad/sample) UsingM = 18, Kaiser window,β = 7 andω 0 = 2.2 = 2π 140kHz 400 khz : Near perfect differentiation forω 1.6 ( 100kHz forf s = 400kHz) Broad transition region allows shorter filter DSP and Digital Filters (2015-7310) FM Radio: 14 9 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. 20kHz j2πn (1) shift spectrum down by20khz: multiply bye 400 khz DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. (1) shift spectrum down by20khz: multiply bye j2πn 20kHz 400 khz (2) low pass filter to±1khz to extract complex pilot at 1kHz: H(z) DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. 20kHz j2πn (1) shift spectrum down by20khz: multiply bye 400 khz (2) low pass filter to±1khz to extract complex pilot at 1kHz: H(z) (3) square to double frequency to 2kHz [ ( e jωt) 2 = e j2ωt ] DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. 20kHz j2πn (1) shift spectrum down by20khz: multiply bye 400 khz (2) low pass filter to±1khz to extract complex pilot at 1kHz: H(z) (3) square to double frequency to 2kHz [ ( e jωt) 2 = e j2ωt ] 40kHz +j2πn (4) shift spectrum up by40khz: multiply bye 400 khz DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. 20kHz j2πn (1) shift spectrum down by20khz: multiply bye 400 khz (2) low pass filter to±1khz to extract complex pilot at 1kHz: H(z) (3) square to double frequency to 2kHz [ ( e jωt) 2 = e j2ωt ] (4) shift spectrum up by40khz: multiply bye +j2πn 40kHz 400 khz (5) take real part DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. 20kHz j2πn (1) shift spectrum down by20khz: multiply bye 400 khz (2) low pass filter to±1khz to extract complex pilot at 1kHz: H(z) (3) square to double frequency to 2kHz [ ( e jωt) 2 = e j2ωt ] 40kHz +j2πn (4) shift spectrum up by40khz: multiply bye 400 khz (5) take real part More efficient to do low pass filtering at a low sample rate: DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. 20kHz j2πn (1) shift spectrum down by20khz: multiply bye 400 khz (2) low pass filter to±1khz to extract complex pilot at 1kHz: H(z) (3) square to double frequency to 2kHz [ ( e jωt) 2 = e j2ωt ] 40kHz +j2πn (4) shift spectrum up by40khz: multiply bye 400 khz (5) take real part More efficient to do low pass filtering at a low sample rate: Transition bands: F(z): 1 19kHz, H(z): 1 3kHz DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. 20kHz j2πn (1) shift spectrum down by20khz: multiply bye 400 khz (2) low pass filter to±1khz to extract complex pilot at 1kHz: H(z) (3) square to double frequency to 2kHz [ ( e jωt) 2 = e j2ωt ] 40kHz +j2πn (4) shift spectrum up by40khz: multiply bye 400 khz (5) take real part More efficient to do low pass filtering at a low sample rate: Transition bands: F(z): 1 19kHz, H(z): 1 3kHz, G(z): 2 18kHz DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Pilot tone extraction (1) (2) Aim: extract19khz pilot tone, double freq real38khz tone. 20kHz j2πn (1) shift spectrum down by20khz: multiply bye 400 khz (2) low pass filter to±1khz to extract complex pilot at 1kHz: H(z) (3) square to double frequency to 2kHz [ ( e jωt) 2 = e j2ωt ] 40kHz +j2πn (4) shift spectrum up by40khz: multiply bye 400 khz (5) take real part More efficient to do low pass filtering at a low sample rate: Transition bands: F(z): 1 19kHz, H(z): 1 3kHz, G(z): 2 18kHz ω = 0.28 M = 60, ω = 0.63 27, ω = 0.25 68 DSP and Digital Filters (2015-7310) FM Radio: 14 10 / 12
Polyphase Pilot tone (1) (2) DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Polyphase Pilot tone (1) (2) DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Polyphase Pilot tone (1) (2) Anti-alias filter: F(z) Each branch,f p (z), gets every20 th sample and an identicale j2π n 20 DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Polyphase Pilot tone (1) (2) Anti-alias filter: F(z) Each branch,f p (z), gets every20 th sample and an identicale j2π n 20 SoF p (z) can filter a real signal and then multiply by fixede j2π p 20 DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Polyphase Pilot tone (1) (2) Anti-alias filter: F(z) Each branch,f p (z), gets every20 th sample and an identicale j2π n 20 SoF p (z) can filter a real signal and then multiply by fixede j2π p 20 Anti-image filter: G(z) Each branch,g p (z), multiplied by identicale j2π n 10 DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Polyphase Pilot tone (1) (2) Anti-alias filter: F(z) Each branch,f p (z), gets every20 th sample and an identicale j2π n 20 SoF p (z) can filter a real signal and then multiply by fixede j2π p 20 Anti-image filter: G(z) Each branch,g p (z), multiplied by identicale j2π n 10 SoG p (z) can filter a real signal DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Polyphase Pilot tone (1) (2) Anti-alias filter: F(z) Each branch,f p (z), gets every20 th sample and an identicale j2π n 20 SoF p (z) can filter a real signal and then multiply by fixede j2π p 20 Anti-image filter: G(z) Each branch,g p (z), multiplied by identicale j2π n 10 SoG p (z) can filter a real signal Multiplies: F andgeach: (3+2) 400kHz DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Polyphase Pilot tone (1) (2) Anti-alias filter: F(z) Each branch,f p (z), gets every20 th sample and an identicale j2π n 20 SoF p (z) can filter a real signal and then multiply by fixede j2π p 20 Anti-image filter: G(z) Each branch,g p (z), multiplied by identicale j2π n 10 SoG p (z) can filter a real signal Multiplies: F andgeach: (3+2) 400kHz,H +x 2 : (2 30+4) 20kHz DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Polyphase Pilot tone (1) (2) Anti-alias filter: F(z) Each branch,f p (z), gets every20 th sample and an identicale j2π n 20 SoF p (z) can filter a real signal and then multiply by fixede j2π p 20 Anti-image filter: G(z) Each branch,g p (z), multiplied by identicale j2π n 10 SoG p (z) can filter a real signal Multiplies: F andgeach: (3+2) 400kHz,H +x 2 : (2 30+4) 20kHz Total: 13.2 400kHz DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Polyphase Pilot tone (1) (2) Anti-alias filter: F(z) Each branch,f p (z), gets every20 th sample and an identicale j2π n 20 SoF p (z) can filter a real signal and then multiply by fixede j2π p 20 Anti-image filter: G(z) Each branch,g p (z), multiplied by identicale j2π n 10 SoG p (z) can filter a real signal Multiplies: F andgeach: (3+2) 400kHz,H +x 2 : (2 30+4) 20kHz Total: 13.2 400kHz [Full-rateH(z) needs 273 400kHz] DSP and Digital Filters (2015-7310) FM Radio: 14 11 / 12
Summary (1) (2) allows sampling below the Nyquist frequency Only works because the wanted signal fits entirely within a Nyquist band image DSP and Digital Filters (2015-7310) FM Radio: 14 12 / 12
Summary (1) (2) allows sampling below the Nyquist frequency Only works because the wanted signal fits entirely within a Nyquist band image Polyphase filter can be combined with complex multiplications to select the desired image subsequent multiplication by j ln shifts by the desired multiple of 1 4 sample rate No actual multiplications required DSP and Digital Filters (2015-7310) FM Radio: 14 12 / 12
Summary (1) (2) allows sampling below the Nyquist frequency Only works because the wanted signal fits entirely within a Nyquist band image Polyphase filter can be combined with complex multiplications to select the desired image subsequent multiplication by j ln shifts by the desired multiple of 1 4 sample rate No actual multiplications required FM demodulation uses a differentiation filter to calculate dφ dt DSP and Digital Filters (2015-7310) FM Radio: 14 12 / 12
Summary (1) (2) allows sampling below the Nyquist frequency Only works because the wanted signal fits entirely within a Nyquist band image Polyphase filter can be combined with complex multiplications to select the desired image subsequent multiplication by j ln shifts by the desired multiple of 1 4 sample rate No actual multiplications required FM demodulation uses a differentiation filter to calculate dφ dt Pilot tone bandpass filter has narrow bandwidth so better done at a low sample rate double the frequency of a complex tone by squaring it DSP and Digital Filters (2015-7310) FM Radio: 14 12 / 12
Summary (1) (2) allows sampling below the Nyquist frequency Only works because the wanted signal fits entirely within a Nyquist band image Polyphase filter can be combined with complex multiplications to select the desired image subsequent multiplication by j ln shifts by the desired multiple of 1 4 sample rate No actual multiplications required FM demodulation uses a differentiation filter to calculate dφ dt Pilot tone bandpass filter has narrow bandwidth so better done at a low sample rate double the frequency of a complex tone by squaring it This example is taken from Harris: 13. DSP and Digital Filters (2015-7310) FM Radio: 14 12 / 12