I've been using the PPS signal to measure oscillator frequency. I feed it into the input capture pin so I know to within a clock cycle when PPS goes high. The number of clocks between successive PPS pulses allows me to calculate the Uno's resonator frequency and I can use that in turn to calibrate against an addition signal. As long as the temperature isn't varying too quickly this works pretty well.
The GPS I'm using has an internal battery backed RTC which maintains the time quite well during relatively brief periods when the GPS loses fix. But the PPS signal goes away when there's no fix; I presume that's the sole reason for adding an external RTC. Rather than trying to synchronize the RTC square wave output to the PPS signal I think it makes more sense to simply determine the offset. You wouldn't even have to actually set the time/date in the RTC since the GPS provides that.
Here's a simple sketch I wrote to measure the frequency of the Uno's resonator. Maybe it will help you. To use this sketch just connect PPS to digital pin 8 (and also ground between the GPS and Uno) and the sketch measures the resonator frequency and error in parts per million, once each second.
// Check processor clock accuracy against GPS
//
// Connect the GPS to the Uno via 5V, GND.
// Connect the PPS signal to the input capture pin D8.
#define PPS_INPUT_PIN 8 // 328 pin 14 (DIP), pin 12 (TQFP) (input capture)
// Pin operation macros
#define INPUT_PORT(pin) (pin < 8 ? PIND : (pin < 14 ? PINB : PINC))
#define OUTPUT_PORT(pin) (pin < 8 ? PORTD : (pin < 14 ? PORTB : PORTC))
#define PIN_MASK(pin) (pin < 8 ? 1<<pin : (pin < 14 ? 1<<pin-8 : 1<<pin-14))
#define SET_PIN(pin, level) (level == LOW ? (OUTPUT_PORT(pin) &= ~PIN_MASK(pin)) : (OUTPUT_PORT(pin) |= PIN_MASK(pin)))
volatile uint16_t captureTicks = 0;
volatile uint16_t captureRolls = 0;
volatile uint16_t rollovers = 0;
volatile bool captureFlag = false;
uint32_t prevCaptureTime = 0;
// =======================================================================================
ISR (TIMER1_CAPT_vect)
{
captureTicks = ICR1;
captureRolls = rollovers;
captureFlag = true;
}
ISR (TIMER1_OVF_vect)
{
rollovers++;
}
// =========================================================================================
void setup()
{
Serial.begin(115200);
Serial.println("\nHello\n");
pinMode(PPS_INPUT_PIN, INPUT);
// Set up timer 1 for input capture of PPS signal
TCCR1A = 0; // reset timer1
TCCR1B = 0;
TCNT1 = 0;
TCCR1B = bit(ICES1) | bit(CS10); // no prescale; capture on rising edge
TIMSK1 = bit(ICIE1) | bit(TOIE1); // enable capture interrupt; enable overflow interrupt
}
void loop()
{
if (captureFlag) {
uint16_t t = captureTicks;
uint16_t r = captureRolls;
captureFlag = false;
uint32_t captureTime = (uint32_t(r) << 16) + uint32_t(t);
uint32_t elapsed = captureTime - prevCaptureTime;
prevCaptureTime = captureTime;
Serial.print(elapsed);
Serial.print(" cycles/sec, ");
Serial.print(float(elapsed)/16.0 - 1000000.0, 1);
Serial.print(" ppm");
Serial.println();
}
}