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ECE387-无代写
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REAL-TIME DIGITAL SYSTEMS DESIGN AND VERIFICATION WITH FPGAS
ECE 387 – LECTURE 13
PROF. DAVID ZARETSKY
DAVID.ZARETSKY@NORTHWESTERN.EDU
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 1
AGENDA
¡ Digital Signal Processing
¡ FM Radio
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 2
DIGITAL SIGNAL PROCESSING BASICS
¡ A basic DSP system is composed of:
¡ An ADC providing digital samples of an analog input
¡ A Digital Processing system (μP/ASIC/FPGA)
¡ A DAC converting processed samples to analog output
¡ Real-time signal processing: All processing operation must be complete between two consecutive samples
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 3
TIME AND FREQUENCY DOMAINS
¡ The time-domain representation gives the amplitudes of signals at the instants of time during which it was
sampled.
¡ Fourier's theorem states that any waveform in the time domain can be represented by the weighted sum of sines
and cosines.
¡ The same waveform then can be represented in the frequency domain as a pair of amplitude and phase values at
each component frequency.
¡ You can generate any waveform by adding sine waves, each with a particular amplitude and phase.
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 4
THE FREQUENCY DOMAIN
¡ The frequency domain does not carry any information that is not in the time domain.
¡ The power in the frequency domain is that it is simply another way of looking at signal information.
¡ Any operation or inspection done in one domain is equally applicable to the other domain, except that usually
one domain makes a particular operation or inspection much easier than in the other domain.
¡ Frequency domain information is extremely important and useful in signal processing.
512/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT
FREQUENCY VS TIME DOMAIN
¡ The figure shows single frequency components spread
out in the time domain, as distinct impulses in the
frequency domain.
¡ The amplitude of each frequency line is the amplitude
of the time waveform for that frequency component.
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 6
THE FOURIER SERIES
¡ Ck is frequency domain amplitude and phase representation
¡ For the given value xp(t) (a square value), the sum of the first four terms of trigonometric Fourier series are:
¡ xp(t) » 1.0 + sin(t) + C2sin(3t) + C3sin(5t)
7
dte)t(x
T
1c
where,ec)t(x
p
0
0
T
tjk
p
p
k
k
tjk
kp
ò
å
-
¥
-¥=
=
=
w
w
Complex Numbers !
Periodic signal expressed as infinite sum of sinusoids.
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT
DIGITAL FILTERING
¡ Filters
¡ Remove unwanted parts of the signal, such as random noise
¡ Extract useful parts of the signal, such as the components lying within a certain frequency range
¡ Analog Filters
¡ Input: electrical voltage or current which is the direct analogue of a physical quantity (sensor output)
¡ Components: resistors, capacitors and op amps
¡ Output: Filtered electrical voltage or current
¡ Applications: noise reduction, video signal enhancement, graphic equalisers
¡ Digital Filters
¡ Input: Digitized samples of analog input (requires ADC)
¡ Components: Digital processor (PC/DSP/ASIC/FPGA)
¡ Output: Filtered samples (requires DAC)
¡ Applications: noise reduction, video signal enhancement, graphic equalisers
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 8
FAST FOURIER TRANSFORM (FFT)
¡ FFT is a digital implementation of the Fourier transform.
¡ FFT resolves a time waveform into its sinusoidal components.
¡ Converts time-domain data into the frequency spectrum of the data.
¡ FFT returns a discrete spectrum, in which the frequency content of the waveform is resolved into a finite number of
frequency lines, or bins.
¡ FFT is a faster version of the Discrete Fourier Transform (DFT)
¡ It utilizes some clever algorithms to do the same thing as the DTF, but in much less time.
¡ Without a discrete-time to discrete-frequency transform we would not be able to compute the Fourier transform with a
microprocessor or FPGA
¡ Use cases for Fourier Transform:
¡ Analyze the frequency spectrum of audio data
¡ Find the frequency components of a signal buried in noise
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 9
FINITE IMPULSE RESPONSE (FIR) FILTERS
¡ FIR filters use past inputs to calculate new output
¡ y(n) = h(0)*x(n) + h(1)*x(n-1) + h(2)*x(n-2) + h(3)*x(n-3)
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT10
FIR SOFTWARE IMPLEMENTATION
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT11
int yn=0; //filter output initialization
short xdly[N+1]; //input delay samples array
void fir()
{
short i;
yn=0;
short h[N] = { //coefficients };
xdly[0] = input_sample();
for (i=0; i yn += (h[i]*xdly[i]);
for (i=N-1; i>0; i--)
xdly[i] = xdly[i-1];
output_sample(yn >> 15);
}
FIR HARDWARE IMPLEMENTATION IN VHDL
entity my_fir is
port (clk, rst: in std_logic;
sample_in: in std_logic_vector(length-1 downto 0);
sample_out: out std_logic_vector(length-1 downto 0)
);
end entity my_fir;
architecture rtl of my_fir is
type taps is array 0 to 3 of std_logic_vector(length-1 downto 0);
constant h : taps := (…); -- coeffients
signal x : taps; --past samples
signal y: std_logic_vector(2*length-1 downto 0);
begin
fir_process : process(x)
variable y_tmp := std_logic_vector(2*length-1 downto 0);
begin
y_tmp := (others => ‘0’);
for i in 0 to length-1 loop
y_tmp := std_logic_vector(signed(h(i)) * signed(x(i)));
end loop;
y <= y_tmp;
end process;
clock_process : process (clk, rst)
begin
if rst=‘1’ then
x <= (others => (others => ‘0’));
elsif rising_edge(clk) then
for i in length-1 downto 1 loop
x(i) <= x(i-1); -- shift
end loop;
x(0) <= sample_in; -- new sample
sample_out <= y(2*length-1 downto length);
end if;
end process;
end architecture rtl;
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT12
INFINITE IMPULSE RESPONSE (IIR) FILTERS
¡ IIR filters use past inputs and outputs to calculate new output
¡ y(n) = b(0)*x(n) + b(1)*x(n-1) + b(2)*x(n-2) + b(3)*x(n-3)
+ a(0)*y(n) + a(1)*y(n-1) + a(2)*y(n-2) + a(3)*y(n-3)
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 13
IIR SOFTWARE IMPLEMENTATION
int yn=0; //filter output initialization
short xdly[N+1]; //input delay samples array
short ydly[M]; //output delay array
void iir()
{
short i;
yn=0;
short a[N] = { //coefficients };
short b[M] = { //coefficients };
xdly[0]=input_sample();
for (i=0; iyn += (b[i]*xdly[i]);
for (i=0; iyn += (a[i]*ydly[i]);
for (i=N-1; i>0; i--)
xdly[i] = xdly[i-1];
ydly[0] = yn >> 15;
for (i=M-1; i>0; i--)
ydly[i] = ydly[i-1];
output_sample(yn >> 15);
}
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT14
FM STEREO RADIO
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 15
FM FREQUENCY SPECTRUM
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 16
93.1
WXRT
94.7
WLS
95.5
WEBG
97.1
WDRV
98.7
WFMT
100.3
WSHE
93.9
WLIT
97.9
WLUP 99.5
WUSN
96.3
WBBM
Fast Fourier Transform (FFT)
8M samples / sec
4096 Bins
DEMODULATING SIGNALS
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 17
INTRODUCTION TO FM RECEIVERS
¡ f(t) = k * m(t) + fc
¡ m(t): the input signal
¡ k: constant that controls the frequency sensitivity
¡ fc : the frequency of the carrier
¡ To recover m(t), two steps are required:
¡ Remove the carrier fc
¡ Compute the instantaneous frequency of the baseband signal
¡ Left & Right audio channels encoded as (L+R) and (L-R)
¡ L+R Mono Channel at fc
¡ L-R Channel at fc + 38 kHz
¡ Stereo Pilot tone at fc + 19 kHz
¡ Total 100 kHz spread
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 18
FM DEMODULATION
¡ Quadrature demodulation produces 2 baseband waveforms that convey the
information that was encoded into the carrier of the received signal.
¡ I and Q waveforms are equivalent to the real and imaginary parts of a complex
number, and are 90-degrees out of phase
¡ Separating I and Q in this way allows you to measure the relative phase of the
components of the signal.
¡ The baseband waveform contained in the modulated signal corresponds to a
magnitude+phase representation of I and Q signals.
¡ The magnitude is M = √(I2 + Q2)
¡ The angle of the I/Q data is ϕ=arctan(Q/I)
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 19
FM RADIO: DECONSTRUCTED
¡ Channel Filter
¡ 20-tap FIR Complex Filter
¡ Cuts off all frequencies above 80 kHz
¡ Demodulator
¡ Differentiates freq by finding diff in angle of phase between consecutive I/Q samples
¡ demod = k * atan( IQ1 * conj(IQ0) )
¡ k is the demodulator gain
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 20
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 21
¡ L+R Channel Filter
¡ Low-Pass 32-tap decimation FIR filter (decimation = 10)
¡ Reduces sampling rate from 320 kHz to 32 kHz
¡ Filters frequencies above 16 kHz
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 22
¡ Stereo Pilot Tone
¡ Identifies whether a stereo signal exists
¡ Band pass 32-tap FIR filter extracts the 19kHz pilot tone
¡ The signal is squared to obtain a 38 kHz cosine
¡ A high pass filter removes the tone at 0Hz created after the pilot
tone is squared
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 23
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
¡ L-R Channel Filter
¡ Band-pass 32-tap FIR filter
¡ Extracts the L-R (23-53 kHz)
sub-carrier frequencies
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 24
¡ L-R Channel Filter
¡ L-R channel is multiplied by the squared pilot signal
¡ L-R channel is demodulate from 38kHz to baseband
¡ Low-Pass decimation FIR reduces
sampling rate (decimation = 10)
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 25
¡ Left & Right Channel Reconstruction
¡ Left Channel: (L+R) + (L-R) = 2L
¡ Right Channel: (L+R) – (L-R) = 2R
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 26
¡ De-emphasis
¡ First-Order 2-Tap IIR Filter improves the signal-to-noise ratio (SNR)
¡ Uses the transfer function H(s) = 1 / (1+s) for RC time circuit and bilinear z-transform
to obtain coefficients (t = 75 us)
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 27
¡ Volume / Gain
¡ Multiply the signal by volume control to increase signal strength
¡ Left/Right channels maintain the same volume control
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
STREAMING DESIGN CONSIDERATIONS
¡ Complex streaming architecture
¡ Has multiple delay paths
¡ How do you remove bottlenecks to ensure data pipeline flows smoothly?
¡ Data quantization to eliminate round-off errors
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 28
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
I/Q
READ
I/Q
Interleaved I/Q data needs
to be partitioned into I
and Q data sets
Data copied along
3 paths
Data copied along
2 paths
Data copied along
2 paths
Data copied along
2 paths
FM STEREO RADIO IN SOFTWARE
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 29
int main(int argc, char **argv)
{
static unsigned char IQ[SAMPLES*4];
static int left_audio[AUDIO_SAMPLES];
static int right_audio[AUDIO_SAMPLES];
if ( argc < 2 )
{
printf("Missing input file.\n");
return -1;
}
// initialize the audio output
int audio_fd = audio_init( AUDIO_RATE );
if ( audio_fd < 0 )
{
printf("Failed to initialize audio!\n");
return -1;
}
FILE * usrp_file = fopen(argv[1], "rb");
if ( usrp_file == NULL ) {
printf("Unable to open file.\n");
return -1;
}
// run the FM receiver
while( !feof(usrp_file) )
{
fread( IQ, sizeof(char), SAMPLES*4, usrp_file );
fm_radio_stereo( IQ, left_audio, right_audio );
audio_tx( audio_fd, AUDIO_RATE, left_audio,
right_audio, AUDIO_SAMPLES );
}
fclose( usrp_file );
close( audio_fd );
return 0;
}
Move to Hardware
FM STEREO RADIO IN SOFTWARE
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 30
// read the I/Q data from the buffer
read_IQ( IQ, I, Q, SAMPLES );
// Channel low-pass filter cuts off all frequnties above 80 Khz
fir_cmplx_n( I, Q, SAMPLES, CHANNEL_COEFFS_REAL,
CHANNEL_COEFFS_IMAG, fir_cmplx_x_real, fir_cmplx_x_imag,
CHANNEL_COEFF_TAPS, 1, I_fir, Q_fir );
// demodulate
demodulate_n( I_fir, Q_fir, demod_real, demod_imag, SAMPLES,
FM_DEMOD_GAIN, demod );
// L+R low-pass FIR - reduce sampling rate from 256 KHz to 32
KHz
fir_n( demod, SAMPLES, AUDIO_LPR_COEFFS, fir_lpr_x,
AUDIO_LPR_COEFF_TAPS, AUDIO_DECIM, audio_lpr_filter );
// L-R band-pass extracts the L-R channel from 23kHz to 53kHz
fir_n( demod, SAMPLES, BP_LMR_COEFFS, fir_bp_x,
BP_LMR_COEFF_TAPS, 1, bp_lmr_filter );
// Pilot band-pass filter extracts the 19kHz pilot tone
fir_n( demod, SAMPLES, BP_PILOT_COEFFS, fir_pilot_x,
BP_PILOT_COEFF_TAPS, 1, bp_pilot_filter );
// square the pilot tone to get 38kHz
multiply_n( bp_pilot_filter, bp_pilot_filter, SAMPLES, square );
// high-pass removes the tone at 0Hz after pilot tone is squared
fir_n( square, SAMPLES, HP_COEFFS, fir_hp_x, HP_COEFF_TAPS, 1,
hp_pilot_filter );
// demodulate the L-R channel from 38kHz to baseband
multiply_n( hp_pilot_filter, bp_lmr_filter, SAMPLES, multiply );
// L-R low-pass FIR - reduce sampling rate from 256 KHz to 32
KHz
fir_n( multiply, SAMPLES, AUDIO_LMR_COEFFS, fir_lmr_x,
AUDIO_LMR_COEFF_TAPS, AUDIO_DECIM, audio_lmr_filter );
// Left audio channel - (L+R) + (L-R) = 2L
add_n( audio_lpr_filter, audio_lmr_filter, AUDIO_SAMPLES, left
);
// Right audio channel - (L+R) - (L-R) = 2R
sub_n( audio_lpr_filter, audio_lmr_filter, AUDIO_SAMPLES, right
);
// Left channel deemphasis
deemphasis_n( left, deemph_l_x, deemph_l_y, AUDIO_SAMPLES,
left_deemph );
// Right channel deemphasis
deemphasis_n( right, deemph_r_x, deemph_r_y, AUDIO_SAMPLES,
right_deemph);
// Left volume control
gain_n( left_deemph, AUDIO_SAMPLES, VOLUME_LEVEL, left_audio );
// Right volume control
gain_n( right_deemph, AUDIO_SAMPLES, VOLUME_LEVEL, right_audio
);
}
void fm_radio_stereo(unsigned char *IQ, int
*left_audio, int *right_audio)
{
static int I[SAMPLES];
static int Q[SAMPLES];
static int I_fir[SAMPLES];
static int Q_fir[SAMPLES];
static int demod[SAMPLES];
static int bp_pilot_filter[SAMPLES];
static int bp_lmr_filter[SAMPLES];
static int hp_pilot_filter[SAMPLES];
static int audio_lpr_filter[AUDIO_SAMPLES];
static int audio_lmr_filter[AUDIO_SAMPLES];
static int square[SAMPLES];
static int multiply[SAMPLES];
static int left[AUDIO_SAMPLES];
static int right[AUDIO_SAMPLES];
static int left_deemph[AUDIO_SAMPLES];
static int right_deemph[AUDIO_SAMPLES];
static int fir_cmplx_x_real[MAX_TAPS];
static int fir_cmplx_x_imag[MAX_TAPS];
static int demod_real[] = {0};
static int demod_imag[] = {0};
static int fir_lpr_x[MAX_TAPS];
static int fir_lmr_x[MAX_TAPS];
static int fir_bp_x[MAX_TAPS];
static int fir_pilot_x[MAX_TAPS];
static int fir_hp_x[MAX_TAPS];
static int deemph_l_x[MAX_TAPS];
static int deemph_l_y[MAX_TAPS];
static int deemph_r_x[MAX_TAPS];
static int deemph_r_y[MAX_TAPS];
SIMPLE FM RADIO FUNCTIONS
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 31
¡ Read I/Q
void read_IQ( unsigned char *IQ, int *I, int *Q, int samples )
{
for ( int i = 0; i < samples; i++ )
{
I[i] = QUANTIZE_I((short)(IQ[i*4+1] << 8) | (short)IQ[i*4+0]);
Q[i] = QUANTIZE_I((short)(IQ[i*4+3] << 8) | (short)IQ[i*4+2]);
}
}
¡ Multiplication
void multiply_n( int *x_in, int *y_in, const int n_samples, int *output )
{
for ( int i = 0; i < n_samples; i++ )
{
output[i] = DEQUANTIZE( x_in[i] * y_in[i] );
}
}
¡ Addition / Subtraction
void add_n( int *x_in, int *y_in, const int n_samples, int *output )
{
for ( int i = 0; i < n_samples; i++ )
{
output[i] = x_in[i] + y_in[i];
}
}
¡ Volume / Gain
void gain_n( int *input, const int n_samples, int gain, int *output )
{
for ( int i = 0; i < n_samples; i++ )
{
output[i] = DEQUANTIZE(input[i] * gain) << (14-BITS);
}
}
DEMODULATION
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 32
#define QUANT_VAL (1 << 10)
#define QUANTIZE_F(f) (int)(((float)(f) * (float)QUANT_VAL))
#define QUANTIZE_I(i) (int)((int)(i) * (int)QUANT_VAL)
#define DEQUANTIZE(i) (int)((int)(i) / (int)QUANT_VAL)
void demod_n( int *real, int *imag, int *real_prev, int *imag_prev,
const int n_samples, const int gain, int *demod_out )
{
for ( int i = 0; i < n_samples; i++ ) {
demodulate( real[i], imag[i], real_prev, imag_prev,
gain, &demod_out[i] );
}
}
void demodulate( int real, int imag, int *real_prev, int *imag_prev,
const int gain, int *demod_out )
{
// k * atan(c1 * conj(c0))
int r = DEQUANTIZE(*real_prev * real) –
DEQUANTIZE(-*imag_prev * imag);
int i = DEQUANTIZE(*real_prev * imag) +
DEQUANTIZE(-*imag_prev * real);
*demod_out = DEQUANTIZE(gain * qarctan(i, r));
*real_prev = real;
*imag_prev = imag;
}
int qarctan(int y, int x)
{
const int quad1 = QUANTIZE_F(PI / 4.0);
const int quad3 = QUANTIZE_F(3.0 * PI / 4.0);
int abs_y = abs(y) + 1;
int angle = 0;
int r = 0;
if ( x >= 0 )
{
r = QUANTIZE_I(x - abs_y) / (x + abs_y);
angle = quad1 - DEQUANTIZE(quad1 * r);
}
else
{
r = QUANTIZE_I(x + abs_y) / (abs_y - x);
angle = quad3 - DEQUANTIZE(quad1 * r);
}
// negate if in quad III or IV
return ((y < 0) ? -angle : angle);
}
Division by a
variable
Streaming input & output => FIFO
FIR DECIMATION FILTER
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 33
#define BITS 10
#define QUANT_VAL (1 << BITS)
#define QUANTIZE_F(f) (int)(((float)(f) * (float)QUANT_VAL))
#define QUANTIZE_I(i) (int)((int)(i) * (int)QUANT_VAL)
#define DEQUANTIZE(i) (int)((int)(i) / (int)QUANT_VAL)
void fir_n( int *x_in, const int n_samples, const int *coeff,
int *x, const int taps, const int decimation,
int *y_out )
{
int i = 0;
int j = 0;
int n_elements = n_samples / decimation;
for ( i = 0; i < n_elements; i++, j+=decimation )
{
fir( &x_in[j], coeff, x, taps, decimation, &y_out[i] );
}
}
void fir( int *x_in, const int *coeff, int *x, const int taps,
const int decimation, int *y_out )
{
int i = 0, j = 0, y = 0;
for ( j = taps-1; j > decimation-1; j-- )
{
x[j] = x[j-decimation];
}
for ( i = 0; i < decimation; i++ )
{
x[decimation-i-1] = x_in[i];
}
for ( j = 0; j < taps; j++ )
{
y += DEQUANTIZE( coeff[taps-j-1] * x[j] );
}
*y_out = y;
}
Streaming input & output => FIFO
FIR COMPLEX FILTER
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 34
#define BITS 10
#define QUANT_VAL (1 << BITS)
#define QUANTIZE_F(f) (int)(((float)(f) * (float)QUANT_VAL))
#define QUANTIZE_I(i) (int)((int)(i) * (int)QUANT_VAL)
#define DEQUANTIZE(i) (int)((int)(i) / (int)QUANT_VAL)
void fir_cmplx_n( int *x_real_in, int *x_imag_in,
const int n_samples, const int *h_real, const int *h_imag,
int *x_real, int *x_imag, const int taps, const int decimation,
int *y_real_out, int *y_imag_out )
{
int i = 0, j = 0;
int n_elements = n_samples / decimation;
for ( ; i < n_elements; i++, j+=decimation )
{
fir_cmplx( &x_real_in[j], &x_imag_in[j], h_real, h_imag, x_real,
x_imag, taps, decimation, &y_real_out[i],
&y_imag_out[i] );
}
}
void fir_cmplx( int *x_real_in, int *x_imag_in, const int *h_real,
const int *h_imag, int *x_real, int *x_imag, const int taps,
const int decimation, int *y_real_out, int *y_imag_out )
{
int i = 0, j = 0, y_real = 0, y_imag = 0;
for ( j = taps-1; j > decimation-1; j-- ) {
x_real[j] = x_real[j-decimation];
x_imag[j] = x_imag[j-decimation];
}
for ( i = 0; i < decimation; i++ ) {
x_real[decimation-i-1] = x_real_in[i];
x_imag[decimation-i-1] = x_imag_in[i];
}
for ( i = 0; i < taps; i++ ) {
y_real += DEQUANTIZE((h_real[i] * x_real[i])
- (h_imag[i] * x_imag[i]));
y_imag += DEQUANTIZE((h_real[i] * x_imag[i])
- (h_imag[i] * x_real[i]));
}
*y_real_out = y_real;
*y_imag_out = y_imag;
}
Streaming input & output => FIFO
DEEMPHASIS / IIR
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 35
#define BITS 10
#define QUANT_VAL (1 << BITS)
#define QUANTIZE_F(f) (int)(((float)(f) * (float)QUANT_VAL))
#define QUANTIZE_I(i) (int)((int)(i) * (int)QUANT_VAL)
#define DEQUANTIZE(i) (int)((int)(i) / (int)QUANT_VAL)
void iir_n( int *x_in, const int n_samples, const int *x_coeffs,
const int *y_coeffs, int *x, int *y, const int taps,
int dec, int *y_out )
{
int i = 0, j = 0;
int n_elements = n_samples / decimation;
for ( ; i < n_elements; i++, j+=decimation )
{
iir(&x_in[j], x_coeffs, y_coeffs, x, y, taps, dec, &y_out[i] );
}
}
void iir( int *x_in, const int *x_coeffs, const int *y_coeffs, int *x,
int *y, const int taps, const int decimation, int *y_out )
{
int y1 = 0, y2 = 0, i = 0, j = 0;
for ( j = taps-1; j > decimation-1; j-- ) {
x[j] = x[j-decimation];
}
for ( i = 0; i < decimation; i++ ) {
x[decimation-i-1] = x_in[i];
}
for ( j = taps-1; j > 0; j--) {
y[j] = y[j-1];
}
for ( i = 0; i < taps; i++ ) {
y1 += DEQUANTIZE( x_coeffs[i] * x[i] );
y2 += DEQUANTIZE( y_coeffs[i] * y[i] );
}
y[0] = y1 + y2;
*y_out = y[taps-1];
}
Streaming input & output => FIFO
NEXT…
¡ Final Project: FM Radio
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 36
ECE 387 – LECTURE 13
PROF. DAVID ZARETSKY
DAVID.ZARETSKY@NORTHWESTERN.EDU
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 1
AGENDA
¡ Digital Signal Processing
¡ FM Radio
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 2
DIGITAL SIGNAL PROCESSING BASICS
¡ A basic DSP system is composed of:
¡ An ADC providing digital samples of an analog input
¡ A Digital Processing system (μP/ASIC/FPGA)
¡ A DAC converting processed samples to analog output
¡ Real-time signal processing: All processing operation must be complete between two consecutive samples
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 3
TIME AND FREQUENCY DOMAINS
¡ The time-domain representation gives the amplitudes of signals at the instants of time during which it was
sampled.
¡ Fourier's theorem states that any waveform in the time domain can be represented by the weighted sum of sines
and cosines.
¡ The same waveform then can be represented in the frequency domain as a pair of amplitude and phase values at
each component frequency.
¡ You can generate any waveform by adding sine waves, each with a particular amplitude and phase.
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 4
THE FREQUENCY DOMAIN
¡ The frequency domain does not carry any information that is not in the time domain.
¡ The power in the frequency domain is that it is simply another way of looking at signal information.
¡ Any operation or inspection done in one domain is equally applicable to the other domain, except that usually
one domain makes a particular operation or inspection much easier than in the other domain.
¡ Frequency domain information is extremely important and useful in signal processing.
512/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT
FREQUENCY VS TIME DOMAIN
¡ The figure shows single frequency components spread
out in the time domain, as distinct impulses in the
frequency domain.
¡ The amplitude of each frequency line is the amplitude
of the time waveform for that frequency component.
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 6
THE FOURIER SERIES
¡ Ck is frequency domain amplitude and phase representation
¡ For the given value xp(t) (a square value), the sum of the first four terms of trigonometric Fourier series are:
¡ xp(t) » 1.0 + sin(t) + C2sin(3t) + C3sin(5t)
7
dte)t(x
T
1c
where,ec)t(x
p
0
0
T
tjk
p
p
k
k
tjk
kp
ò
å
-
¥
-¥=
=
=
w
w
Complex Numbers !
Periodic signal expressed as infinite sum of sinusoids.
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT
DIGITAL FILTERING
¡ Filters
¡ Remove unwanted parts of the signal, such as random noise
¡ Extract useful parts of the signal, such as the components lying within a certain frequency range
¡ Analog Filters
¡ Input: electrical voltage or current which is the direct analogue of a physical quantity (sensor output)
¡ Components: resistors, capacitors and op amps
¡ Output: Filtered electrical voltage or current
¡ Applications: noise reduction, video signal enhancement, graphic equalisers
¡ Digital Filters
¡ Input: Digitized samples of analog input (requires ADC)
¡ Components: Digital processor (PC/DSP/ASIC/FPGA)
¡ Output: Filtered samples (requires DAC)
¡ Applications: noise reduction, video signal enhancement, graphic equalisers
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 8
FAST FOURIER TRANSFORM (FFT)
¡ FFT is a digital implementation of the Fourier transform.
¡ FFT resolves a time waveform into its sinusoidal components.
¡ Converts time-domain data into the frequency spectrum of the data.
¡ FFT returns a discrete spectrum, in which the frequency content of the waveform is resolved into a finite number of
frequency lines, or bins.
¡ FFT is a faster version of the Discrete Fourier Transform (DFT)
¡ It utilizes some clever algorithms to do the same thing as the DTF, but in much less time.
¡ Without a discrete-time to discrete-frequency transform we would not be able to compute the Fourier transform with a
microprocessor or FPGA
¡ Use cases for Fourier Transform:
¡ Analyze the frequency spectrum of audio data
¡ Find the frequency components of a signal buried in noise
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 9
FINITE IMPULSE RESPONSE (FIR) FILTERS
¡ FIR filters use past inputs to calculate new output
¡ y(n) = h(0)*x(n) + h(1)*x(n-1) + h(2)*x(n-2) + h(3)*x(n-3)
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT10
FIR SOFTWARE IMPLEMENTATION
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT11
int yn=0; //filter output initialization
short xdly[N+1]; //input delay samples array
void fir()
{
short i;
yn=0;
short h[N] = { //coefficients };
xdly[0] = input_sample();
for (i=0; i
for (i=N-1; i>0; i--)
xdly[i] = xdly[i-1];
output_sample(yn >> 15);
}
FIR HARDWARE IMPLEMENTATION IN VHDL
entity my_fir is
port (clk, rst: in std_logic;
sample_in: in std_logic_vector(length-1 downto 0);
sample_out: out std_logic_vector(length-1 downto 0)
);
end entity my_fir;
architecture rtl of my_fir is
type taps is array 0 to 3 of std_logic_vector(length-1 downto 0);
constant h : taps := (…); -- coeffients
signal x : taps; --past samples
signal y: std_logic_vector(2*length-1 downto 0);
begin
fir_process : process(x)
variable y_tmp := std_logic_vector(2*length-1 downto 0);
begin
y_tmp := (others => ‘0’);
for i in 0 to length-1 loop
y_tmp := std_logic_vector(signed(h(i)) * signed(x(i)));
end loop;
y <= y_tmp;
end process;
clock_process : process (clk, rst)
begin
if rst=‘1’ then
x <= (others => (others => ‘0’));
elsif rising_edge(clk) then
for i in length-1 downto 1 loop
x(i) <= x(i-1); -- shift
end loop;
x(0) <= sample_in; -- new sample
sample_out <= y(2*length-1 downto length);
end if;
end process;
end architecture rtl;
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT12
INFINITE IMPULSE RESPONSE (IIR) FILTERS
¡ IIR filters use past inputs and outputs to calculate new output
¡ y(n) = b(0)*x(n) + b(1)*x(n-1) + b(2)*x(n-2) + b(3)*x(n-3)
+ a(0)*y(n) + a(1)*y(n-1) + a(2)*y(n-2) + a(3)*y(n-3)
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 13
IIR SOFTWARE IMPLEMENTATION
int yn=0; //filter output initialization
short xdly[N+1]; //input delay samples array
short ydly[M]; //output delay array
void iir()
{
short i;
yn=0;
short a[N] = { //coefficients };
short b[M] = { //coefficients };
xdly[0]=input_sample();
for (i=0; i
for (i=0; i
for (i=N-1; i>0; i--)
xdly[i] = xdly[i-1];
ydly[0] = yn >> 15;
for (i=M-1; i>0; i--)
ydly[i] = ydly[i-1];
output_sample(yn >> 15);
}
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT14
FM STEREO RADIO
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 15
FM FREQUENCY SPECTRUM
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 16
93.1
WXRT
94.7
WLS
95.5
WEBG
97.1
WDRV
98.7
WFMT
100.3
WSHE
93.9
WLIT
97.9
WLUP 99.5
WUSN
96.3
WBBM
Fast Fourier Transform (FFT)
8M samples / sec
4096 Bins
DEMODULATING SIGNALS
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 17
INTRODUCTION TO FM RECEIVERS
¡ f(t) = k * m(t) + fc
¡ m(t): the input signal
¡ k: constant that controls the frequency sensitivity
¡ fc : the frequency of the carrier
¡ To recover m(t), two steps are required:
¡ Remove the carrier fc
¡ Compute the instantaneous frequency of the baseband signal
¡ Left & Right audio channels encoded as (L+R) and (L-R)
¡ L+R Mono Channel at fc
¡ L-R Channel at fc + 38 kHz
¡ Stereo Pilot tone at fc + 19 kHz
¡ Total 100 kHz spread
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 18
FM DEMODULATION
¡ Quadrature demodulation produces 2 baseband waveforms that convey the
information that was encoded into the carrier of the received signal.
¡ I and Q waveforms are equivalent to the real and imaginary parts of a complex
number, and are 90-degrees out of phase
¡ Separating I and Q in this way allows you to measure the relative phase of the
components of the signal.
¡ The baseband waveform contained in the modulated signal corresponds to a
magnitude+phase representation of I and Q signals.
¡ The magnitude is M = √(I2 + Q2)
¡ The angle of the I/Q data is ϕ=arctan(Q/I)
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 19
FM RADIO: DECONSTRUCTED
¡ Channel Filter
¡ 20-tap FIR Complex Filter
¡ Cuts off all frequencies above 80 kHz
¡ Demodulator
¡ Differentiates freq by finding diff in angle of phase between consecutive I/Q samples
¡ demod = k * atan( IQ1 * conj(IQ0) )
¡ k is the demodulator gain
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 20
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 21
¡ L+R Channel Filter
¡ Low-Pass 32-tap decimation FIR filter (decimation = 10)
¡ Reduces sampling rate from 320 kHz to 32 kHz
¡ Filters frequencies above 16 kHz
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 22
¡ Stereo Pilot Tone
¡ Identifies whether a stereo signal exists
¡ Band pass 32-tap FIR filter extracts the 19kHz pilot tone
¡ The signal is squared to obtain a 38 kHz cosine
¡ A high pass filter removes the tone at 0Hz created after the pilot
tone is squared
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 23
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
¡ L-R Channel Filter
¡ Band-pass 32-tap FIR filter
¡ Extracts the L-R (23-53 kHz)
sub-carrier frequencies
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 24
¡ L-R Channel Filter
¡ L-R channel is multiplied by the squared pilot signal
¡ L-R channel is demodulate from 38kHz to baseband
¡ Low-Pass decimation FIR reduces
sampling rate (decimation = 10)
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 25
¡ Left & Right Channel Reconstruction
¡ Left Channel: (L+R) + (L-R) = 2L
¡ Right Channel: (L+R) – (L-R) = 2R
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 26
¡ De-emphasis
¡ First-Order 2-Tap IIR Filter improves the signal-to-noise ratio (SNR)
¡ Uses the transfer function H(s) = 1 / (1+s) for RC time circuit and bilinear z-transform
to obtain coefficients (t = 75 us)
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
FM RADIO: DECONSTRUCTED
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 27
¡ Volume / Gain
¡ Multiply the signal by volume control to increase signal strength
¡ Left/Right channels maintain the same volume control
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Q
I
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
STREAMING DESIGN CONSIDERATIONS
¡ Complex streaming architecture
¡ Has multiple delay paths
¡ How do you remove bottlenecks to ensure data pipeline flows smoothly?
¡ Data quantization to eliminate round-off errors
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 28
Right
Audio
FIR
CMPLX DEMOD FIR DEEMPH GAIN
Left
Audio
Volume
FIR MULT FIR
MULT FIR
FIR
ADD
SUB GAINDEEMPH
I/Q
READ
I/Q
Interleaved I/Q data needs
to be partitioned into I
and Q data sets
Data copied along
3 paths
Data copied along
2 paths
Data copied along
2 paths
Data copied along
2 paths
FM STEREO RADIO IN SOFTWARE
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 29
int main(int argc, char **argv)
{
static unsigned char IQ[SAMPLES*4];
static int left_audio[AUDIO_SAMPLES];
static int right_audio[AUDIO_SAMPLES];
if ( argc < 2 )
{
printf("Missing input file.\n");
return -1;
}
// initialize the audio output
int audio_fd = audio_init( AUDIO_RATE );
if ( audio_fd < 0 )
{
printf("Failed to initialize audio!\n");
return -1;
}
FILE * usrp_file = fopen(argv[1], "rb");
if ( usrp_file == NULL ) {
printf("Unable to open file.\n");
return -1;
}
// run the FM receiver
while( !feof(usrp_file) )
{
fread( IQ, sizeof(char), SAMPLES*4, usrp_file );
fm_radio_stereo( IQ, left_audio, right_audio );
audio_tx( audio_fd, AUDIO_RATE, left_audio,
right_audio, AUDIO_SAMPLES );
}
fclose( usrp_file );
close( audio_fd );
return 0;
}
Move to Hardware
FM STEREO RADIO IN SOFTWARE
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 30
// read the I/Q data from the buffer
read_IQ( IQ, I, Q, SAMPLES );
// Channel low-pass filter cuts off all frequnties above 80 Khz
fir_cmplx_n( I, Q, SAMPLES, CHANNEL_COEFFS_REAL,
CHANNEL_COEFFS_IMAG, fir_cmplx_x_real, fir_cmplx_x_imag,
CHANNEL_COEFF_TAPS, 1, I_fir, Q_fir );
// demodulate
demodulate_n( I_fir, Q_fir, demod_real, demod_imag, SAMPLES,
FM_DEMOD_GAIN, demod );
// L+R low-pass FIR - reduce sampling rate from 256 KHz to 32
KHz
fir_n( demod, SAMPLES, AUDIO_LPR_COEFFS, fir_lpr_x,
AUDIO_LPR_COEFF_TAPS, AUDIO_DECIM, audio_lpr_filter );
// L-R band-pass extracts the L-R channel from 23kHz to 53kHz
fir_n( demod, SAMPLES, BP_LMR_COEFFS, fir_bp_x,
BP_LMR_COEFF_TAPS, 1, bp_lmr_filter );
// Pilot band-pass filter extracts the 19kHz pilot tone
fir_n( demod, SAMPLES, BP_PILOT_COEFFS, fir_pilot_x,
BP_PILOT_COEFF_TAPS, 1, bp_pilot_filter );
// square the pilot tone to get 38kHz
multiply_n( bp_pilot_filter, bp_pilot_filter, SAMPLES, square );
// high-pass removes the tone at 0Hz after pilot tone is squared
fir_n( square, SAMPLES, HP_COEFFS, fir_hp_x, HP_COEFF_TAPS, 1,
hp_pilot_filter );
// demodulate the L-R channel from 38kHz to baseband
multiply_n( hp_pilot_filter, bp_lmr_filter, SAMPLES, multiply );
// L-R low-pass FIR - reduce sampling rate from 256 KHz to 32
KHz
fir_n( multiply, SAMPLES, AUDIO_LMR_COEFFS, fir_lmr_x,
AUDIO_LMR_COEFF_TAPS, AUDIO_DECIM, audio_lmr_filter );
// Left audio channel - (L+R) + (L-R) = 2L
add_n( audio_lpr_filter, audio_lmr_filter, AUDIO_SAMPLES, left
);
// Right audio channel - (L+R) - (L-R) = 2R
sub_n( audio_lpr_filter, audio_lmr_filter, AUDIO_SAMPLES, right
);
// Left channel deemphasis
deemphasis_n( left, deemph_l_x, deemph_l_y, AUDIO_SAMPLES,
left_deemph );
// Right channel deemphasis
deemphasis_n( right, deemph_r_x, deemph_r_y, AUDIO_SAMPLES,
right_deemph);
// Left volume control
gain_n( left_deemph, AUDIO_SAMPLES, VOLUME_LEVEL, left_audio );
// Right volume control
gain_n( right_deemph, AUDIO_SAMPLES, VOLUME_LEVEL, right_audio
);
}
void fm_radio_stereo(unsigned char *IQ, int
*left_audio, int *right_audio)
{
static int I[SAMPLES];
static int Q[SAMPLES];
static int I_fir[SAMPLES];
static int Q_fir[SAMPLES];
static int demod[SAMPLES];
static int bp_pilot_filter[SAMPLES];
static int bp_lmr_filter[SAMPLES];
static int hp_pilot_filter[SAMPLES];
static int audio_lpr_filter[AUDIO_SAMPLES];
static int audio_lmr_filter[AUDIO_SAMPLES];
static int square[SAMPLES];
static int multiply[SAMPLES];
static int left[AUDIO_SAMPLES];
static int right[AUDIO_SAMPLES];
static int left_deemph[AUDIO_SAMPLES];
static int right_deemph[AUDIO_SAMPLES];
static int fir_cmplx_x_real[MAX_TAPS];
static int fir_cmplx_x_imag[MAX_TAPS];
static int demod_real[] = {0};
static int demod_imag[] = {0};
static int fir_lpr_x[MAX_TAPS];
static int fir_lmr_x[MAX_TAPS];
static int fir_bp_x[MAX_TAPS];
static int fir_pilot_x[MAX_TAPS];
static int fir_hp_x[MAX_TAPS];
static int deemph_l_x[MAX_TAPS];
static int deemph_l_y[MAX_TAPS];
static int deemph_r_x[MAX_TAPS];
static int deemph_r_y[MAX_TAPS];
SIMPLE FM RADIO FUNCTIONS
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 31
¡ Read I/Q
void read_IQ( unsigned char *IQ, int *I, int *Q, int samples )
{
for ( int i = 0; i < samples; i++ )
{
I[i] = QUANTIZE_I((short)(IQ[i*4+1] << 8) | (short)IQ[i*4+0]);
Q[i] = QUANTIZE_I((short)(IQ[i*4+3] << 8) | (short)IQ[i*4+2]);
}
}
¡ Multiplication
void multiply_n( int *x_in, int *y_in, const int n_samples, int *output )
{
for ( int i = 0; i < n_samples; i++ )
{
output[i] = DEQUANTIZE( x_in[i] * y_in[i] );
}
}
¡ Addition / Subtraction
void add_n( int *x_in, int *y_in, const int n_samples, int *output )
{
for ( int i = 0; i < n_samples; i++ )
{
output[i] = x_in[i] + y_in[i];
}
}
¡ Volume / Gain
void gain_n( int *input, const int n_samples, int gain, int *output )
{
for ( int i = 0; i < n_samples; i++ )
{
output[i] = DEQUANTIZE(input[i] * gain) << (14-BITS);
}
}
DEMODULATION
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 32
#define QUANT_VAL (1 << 10)
#define QUANTIZE_F(f) (int)(((float)(f) * (float)QUANT_VAL))
#define QUANTIZE_I(i) (int)((int)(i) * (int)QUANT_VAL)
#define DEQUANTIZE(i) (int)((int)(i) / (int)QUANT_VAL)
void demod_n( int *real, int *imag, int *real_prev, int *imag_prev,
const int n_samples, const int gain, int *demod_out )
{
for ( int i = 0; i < n_samples; i++ ) {
demodulate( real[i], imag[i], real_prev, imag_prev,
gain, &demod_out[i] );
}
}
void demodulate( int real, int imag, int *real_prev, int *imag_prev,
const int gain, int *demod_out )
{
// k * atan(c1 * conj(c0))
int r = DEQUANTIZE(*real_prev * real) –
DEQUANTIZE(-*imag_prev * imag);
int i = DEQUANTIZE(*real_prev * imag) +
DEQUANTIZE(-*imag_prev * real);
*demod_out = DEQUANTIZE(gain * qarctan(i, r));
*real_prev = real;
*imag_prev = imag;
}
int qarctan(int y, int x)
{
const int quad1 = QUANTIZE_F(PI / 4.0);
const int quad3 = QUANTIZE_F(3.0 * PI / 4.0);
int abs_y = abs(y) + 1;
int angle = 0;
int r = 0;
if ( x >= 0 )
{
r = QUANTIZE_I(x - abs_y) / (x + abs_y);
angle = quad1 - DEQUANTIZE(quad1 * r);
}
else
{
r = QUANTIZE_I(x + abs_y) / (abs_y - x);
angle = quad3 - DEQUANTIZE(quad1 * r);
}
// negate if in quad III or IV
return ((y < 0) ? -angle : angle);
}
Division by a
variable
Streaming input & output => FIFO
FIR DECIMATION FILTER
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 33
#define BITS 10
#define QUANT_VAL (1 << BITS)
#define QUANTIZE_F(f) (int)(((float)(f) * (float)QUANT_VAL))
#define QUANTIZE_I(i) (int)((int)(i) * (int)QUANT_VAL)
#define DEQUANTIZE(i) (int)((int)(i) / (int)QUANT_VAL)
void fir_n( int *x_in, const int n_samples, const int *coeff,
int *x, const int taps, const int decimation,
int *y_out )
{
int i = 0;
int j = 0;
int n_elements = n_samples / decimation;
for ( i = 0; i < n_elements; i++, j+=decimation )
{
fir( &x_in[j], coeff, x, taps, decimation, &y_out[i] );
}
}
void fir( int *x_in, const int *coeff, int *x, const int taps,
const int decimation, int *y_out )
{
int i = 0, j = 0, y = 0;
for ( j = taps-1; j > decimation-1; j-- )
{
x[j] = x[j-decimation];
}
for ( i = 0; i < decimation; i++ )
{
x[decimation-i-1] = x_in[i];
}
for ( j = 0; j < taps; j++ )
{
y += DEQUANTIZE( coeff[taps-j-1] * x[j] );
}
*y_out = y;
}
Streaming input & output => FIFO
FIR COMPLEX FILTER
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 34
#define BITS 10
#define QUANT_VAL (1 << BITS)
#define QUANTIZE_F(f) (int)(((float)(f) * (float)QUANT_VAL))
#define QUANTIZE_I(i) (int)((int)(i) * (int)QUANT_VAL)
#define DEQUANTIZE(i) (int)((int)(i) / (int)QUANT_VAL)
void fir_cmplx_n( int *x_real_in, int *x_imag_in,
const int n_samples, const int *h_real, const int *h_imag,
int *x_real, int *x_imag, const int taps, const int decimation,
int *y_real_out, int *y_imag_out )
{
int i = 0, j = 0;
int n_elements = n_samples / decimation;
for ( ; i < n_elements; i++, j+=decimation )
{
fir_cmplx( &x_real_in[j], &x_imag_in[j], h_real, h_imag, x_real,
x_imag, taps, decimation, &y_real_out[i],
&y_imag_out[i] );
}
}
void fir_cmplx( int *x_real_in, int *x_imag_in, const int *h_real,
const int *h_imag, int *x_real, int *x_imag, const int taps,
const int decimation, int *y_real_out, int *y_imag_out )
{
int i = 0, j = 0, y_real = 0, y_imag = 0;
for ( j = taps-1; j > decimation-1; j-- ) {
x_real[j] = x_real[j-decimation];
x_imag[j] = x_imag[j-decimation];
}
for ( i = 0; i < decimation; i++ ) {
x_real[decimation-i-1] = x_real_in[i];
x_imag[decimation-i-1] = x_imag_in[i];
}
for ( i = 0; i < taps; i++ ) {
y_real += DEQUANTIZE((h_real[i] * x_real[i])
- (h_imag[i] * x_imag[i]));
y_imag += DEQUANTIZE((h_real[i] * x_imag[i])
- (h_imag[i] * x_real[i]));
}
*y_real_out = y_real;
*y_imag_out = y_imag;
}
Streaming input & output => FIFO
DEEMPHASIS / IIR
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 35
#define BITS 10
#define QUANT_VAL (1 << BITS)
#define QUANTIZE_F(f) (int)(((float)(f) * (float)QUANT_VAL))
#define QUANTIZE_I(i) (int)((int)(i) * (int)QUANT_VAL)
#define DEQUANTIZE(i) (int)((int)(i) / (int)QUANT_VAL)
void iir_n( int *x_in, const int n_samples, const int *x_coeffs,
const int *y_coeffs, int *x, int *y, const int taps,
int dec, int *y_out )
{
int i = 0, j = 0;
int n_elements = n_samples / decimation;
for ( ; i < n_elements; i++, j+=decimation )
{
iir(&x_in[j], x_coeffs, y_coeffs, x, y, taps, dec, &y_out[i] );
}
}
void iir( int *x_in, const int *x_coeffs, const int *y_coeffs, int *x,
int *y, const int taps, const int decimation, int *y_out )
{
int y1 = 0, y2 = 0, i = 0, j = 0;
for ( j = taps-1; j > decimation-1; j-- ) {
x[j] = x[j-decimation];
}
for ( i = 0; i < decimation; i++ ) {
x[decimation-i-1] = x_in[i];
}
for ( j = taps-1; j > 0; j--) {
y[j] = y[j-1];
}
for ( i = 0; i < taps; i++ ) {
y1 += DEQUANTIZE( x_coeffs[i] * x[i] );
y2 += DEQUANTIZE( y_coeffs[i] * y[i] );
}
y[0] = y1 + y2;
*y_out = y[taps-1];
}
Streaming input & output => FIFO
NEXT…
¡ Final Project: FM Radio
12/25/23NORTHWESTERN UNIVERSITY – ECE DEPARTMENT 36
